set cR = center R;

set ccR = the carrier of (center R);

set ccs = the carrier of R;

set lm = the multF of R | [: the carrier of (center R), the carrier of R:];

A1: the carrier of (center R) c= the carrier of R by Th16;

A2: dom the multF of R = [: the carrier of R, the carrier of R:] by FUNCT_2:def 1;

[: the carrier of (center R), the carrier of R:] c= [: the carrier of R, the carrier of R:]

set Vos = ModuleStr(# the carrier of R, the addF of R,(0. R),lm #);

set cV = the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #);

A8: ( ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is vector-distributive & ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-distributive & ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-associative & ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-unital )_{1} being strict VectSp of center R st

( addLoopStr(# the carrier of b_{1}, the addF of b_{1}, the ZeroF of b_{1} #) = addLoopStr(# the carrier of R, the addF of R, the ZeroF of R #) & the lmult of b_{1} = the multF of R | [: the carrier of (center R), the carrier of R:] )
by A8, A23, A24, A25; :: thesis: verum

set ccR = the carrier of (center R);

set ccs = the carrier of R;

set lm = the multF of R | [: the carrier of (center R), the carrier of R:];

A1: the carrier of (center R) c= the carrier of R by Th16;

A2: dom the multF of R = [: the carrier of R, the carrier of R:] by FUNCT_2:def 1;

[: the carrier of (center R), the carrier of R:] c= [: the carrier of R, the carrier of R:]

proof

then A3:
dom ( the multF of R | [: the carrier of (center R), the carrier of R:]) = [: the carrier of (center R), the carrier of R:]
by A2, RELAT_1:62;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in [: the carrier of (center R), the carrier of R:] or x in [: the carrier of R, the carrier of R:] )

assume x in [: the carrier of (center R), the carrier of R:] ; :: thesis: x in [: the carrier of R, the carrier of R:]

then ex x1, x2 being object st

( x1 in the carrier of (center R) & x2 in the carrier of R & x = [x1,x2] ) by ZFMISC_1:def 2;

hence x in [: the carrier of R, the carrier of R:] by A1, ZFMISC_1:def 2; :: thesis: verum

end;assume x in [: the carrier of (center R), the carrier of R:] ; :: thesis: x in [: the carrier of R, the carrier of R:]

then ex x1, x2 being object st

( x1 in the carrier of (center R) & x2 in the carrier of R & x = [x1,x2] ) by ZFMISC_1:def 2;

hence x in [: the carrier of R, the carrier of R:] by A1, ZFMISC_1:def 2; :: thesis: verum

now :: thesis: for x being object st x in [: the carrier of (center R), the carrier of R:] holds

( the multF of R | [: the carrier of (center R), the carrier of R:]) . x in the carrier of R

then reconsider lm = the multF of R | [: the carrier of (center R), the carrier of R:] as Function of [: the carrier of (center R), the carrier of R:], the carrier of R by A3, FUNCT_2:3;( the multF of R | [: the carrier of (center R), the carrier of R:]) . x in the carrier of R

let x be object ; :: thesis: ( x in [: the carrier of (center R), the carrier of R:] implies ( the multF of R | [: the carrier of (center R), the carrier of R:]) . x in the carrier of R )

assume A4: x in [: the carrier of (center R), the carrier of R:] ; :: thesis: ( the multF of R | [: the carrier of (center R), the carrier of R:]) . x in the carrier of R

then consider x1, x2 being object such that

A5: x1 in the carrier of (center R) and

A6: x2 in the carrier of R and

A7: x = [x1,x2] by ZFMISC_1:def 2;

reconsider x1 = x1 as Element of R by A1, A5;

reconsider x2 = x2 as Element of R by A6;

( the multF of R | [: the carrier of (center R), the carrier of R:]) . x = x1 * x2 by A4, A7, FUNCT_1:49;

hence ( the multF of R | [: the carrier of (center R), the carrier of R:]) . x in the carrier of R ; :: thesis: verum

end;assume A4: x in [: the carrier of (center R), the carrier of R:] ; :: thesis: ( the multF of R | [: the carrier of (center R), the carrier of R:]) . x in the carrier of R

then consider x1, x2 being object such that

A5: x1 in the carrier of (center R) and

A6: x2 in the carrier of R and

A7: x = [x1,x2] by ZFMISC_1:def 2;

reconsider x1 = x1 as Element of R by A1, A5;

reconsider x2 = x2 as Element of R by A6;

( the multF of R | [: the carrier of (center R), the carrier of R:]) . x = x1 * x2 by A4, A7, FUNCT_1:49;

hence ( the multF of R | [: the carrier of (center R), the carrier of R:]) . x in the carrier of R ; :: thesis: verum

set Vos = ModuleStr(# the carrier of R, the addF of R,(0. R),lm #);

set cV = the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #);

A8: ( ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is vector-distributive & ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-distributive & ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-associative & ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-unital )

proof

A23:
ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is add-associative
A9:
the multF of (center R) = the multF of R || the carrier of (center R)
by Def4;

A10: the addF of (center R) = the addF of R || the carrier of (center R) by Def4;

thus ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is vector-distributive :: thesis: ( ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-distributive & ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-associative & ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-unital )

reconsider vv = v as Element of R ;

1_ R in center R by Th19;

then 1_ R in the carrier of (center R) ;

then A22: [(1_ R),vv] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

thus (1. (center R)) * v = lm . ((1. R),vv) by Def4

.= (1. R) * vv by A22, FUNCT_1:49

.= v ; :: thesis: verum

end;A10: the addF of (center R) = the addF of R || the carrier of (center R) by Def4;

thus ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is vector-distributive :: thesis: ( ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-distributive & ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-associative & ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-unital )

proof

thus
ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-distributive
:: thesis: ( ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-associative & ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-unital )
let x be Element of the carrier of (center R); :: according to VECTSP_1:def 13 :: thesis: for b_{1}, b_{2} being Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) holds x * (b_{1} + b_{2}) = (x * b_{1}) + (x * b_{2})

let v, w be Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #); :: thesis: x * (v + w) = (x * v) + (x * w)

reconsider xx = x as Element of R by A1;

reconsider vv = v, ww = w as Element of R ;

A11: [x,w] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A12: [x,(v + w)] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A13: [x,v] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

thus x * (v + w) = xx * (vv + ww) by A12, FUNCT_1:49

.= (xx * vv) + (xx * ww) by VECTSP_1:def 2

.= the addF of R . [(x * v),( the multF of R . [xx,ww])] by A13, FUNCT_1:49

.= (x * v) + (x * w) by A11, FUNCT_1:49 ; :: thesis: verum

end;let v, w be Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #); :: thesis: x * (v + w) = (x * v) + (x * w)

reconsider xx = x as Element of R by A1;

reconsider vv = v, ww = w as Element of R ;

A11: [x,w] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A12: [x,(v + w)] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A13: [x,v] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

thus x * (v + w) = xx * (vv + ww) by A12, FUNCT_1:49

.= (xx * vv) + (xx * ww) by VECTSP_1:def 2

.= the addF of R . [(x * v),( the multF of R . [xx,ww])] by A13, FUNCT_1:49

.= (x * v) + (x * w) by A11, FUNCT_1:49 ; :: thesis: verum

proof

thus
ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-associative
:: thesis: ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is scalar-unital
let x, y be Element of the carrier of (center R); :: according to VECTSP_1:def 14 :: thesis: for b_{1} being Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) holds (x + y) * b_{1} = (x * b_{1}) + (y * b_{1})

let v be Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #); :: thesis: (x + y) * v = (x * v) + (y * v)

reconsider xx = x, yy = y as Element of R by A1;

reconsider vv = v as Element of R ;

A14: [y,v] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A15: [x,v] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A16: [(x + y),v] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A17: [x,y] in [: the carrier of (center R), the carrier of (center R):] by ZFMISC_1:def 2;

thus (x + y) * v = the multF of R . [( the addF of (center R) . [x,y]),vv] by A16, FUNCT_1:49

.= (xx + yy) * vv by A10, A17, FUNCT_1:49

.= (xx * vv) + (yy * vv) by VECTSP_1:def 3

.= the addF of R . [(x * v),( the multF of R . [yy,vv])] by A15, FUNCT_1:49

.= (x * v) + (y * v) by A14, FUNCT_1:49 ; :: thesis: verum

end;let v be Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #); :: thesis: (x + y) * v = (x * v) + (y * v)

reconsider xx = x, yy = y as Element of R by A1;

reconsider vv = v as Element of R ;

A14: [y,v] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A15: [x,v] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A16: [(x + y),v] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A17: [x,y] in [: the carrier of (center R), the carrier of (center R):] by ZFMISC_1:def 2;

thus (x + y) * v = the multF of R . [( the addF of (center R) . [x,y]),vv] by A16, FUNCT_1:49

.= (xx + yy) * vv by A10, A17, FUNCT_1:49

.= (xx * vv) + (yy * vv) by VECTSP_1:def 3

.= the addF of R . [(x * v),( the multF of R . [yy,vv])] by A15, FUNCT_1:49

.= (x * v) + (y * v) by A14, FUNCT_1:49 ; :: thesis: verum

proof

let v be Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #); :: according to VECTSP_1:def 16 :: thesis: (1. (center R)) * v = v
let x, y be Element of the carrier of (center R); :: according to VECTSP_1:def 15 :: thesis: for b_{1} being Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) holds (x * y) * b_{1} = x * (y * b_{1})

let v be Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #); :: thesis: (x * y) * v = x * (y * v)

reconsider xx = x, yy = y as Element of R by A1;

reconsider vv = v as Element of R ;

A18: [x,(y * v)] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A19: [y,v] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A20: [(x * y),v] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A21: [x,y] in [: the carrier of (center R), the carrier of (center R):] by ZFMISC_1:def 2;

thus (x * y) * v = the multF of R . [( the multF of (center R) . (x,y)),vv] by A20, FUNCT_1:49

.= (xx * yy) * vv by A9, A21, FUNCT_1:49

.= xx * (yy * vv) by GROUP_1:def 3

.= the multF of R . [xx,(lm . (y,v))] by A19, FUNCT_1:49

.= x * (y * v) by A18, FUNCT_1:49 ; :: thesis: verum

end;let v be Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #); :: thesis: (x * y) * v = x * (y * v)

reconsider xx = x, yy = y as Element of R by A1;

reconsider vv = v as Element of R ;

A18: [x,(y * v)] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A19: [y,v] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A20: [(x * y),v] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

A21: [x,y] in [: the carrier of (center R), the carrier of (center R):] by ZFMISC_1:def 2;

thus (x * y) * v = the multF of R . [( the multF of (center R) . (x,y)),vv] by A20, FUNCT_1:49

.= (xx * yy) * vv by A9, A21, FUNCT_1:49

.= xx * (yy * vv) by GROUP_1:def 3

.= the multF of R . [xx,(lm . (y,v))] by A19, FUNCT_1:49

.= x * (y * v) by A18, FUNCT_1:49 ; :: thesis: verum

reconsider vv = v as Element of R ;

1_ R in center R by Th19;

then 1_ R in the carrier of (center R) ;

then A22: [(1_ R),vv] in [: the carrier of (center R), the carrier of R:] by ZFMISC_1:def 2;

thus (1. (center R)) * v = lm . ((1. R),vv) by Def4

.= (1. R) * vv by A22, FUNCT_1:49

.= v ; :: thesis: verum

proof

A24:
ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is right_zeroed
let u, v, w be Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #); :: according to RLVECT_1:def 3 :: thesis: (u + v) + w = u + (v + w)

reconsider uu = u, vv = v, ww = w as Element of the carrier of R ;

thus (u + v) + w = (uu + vv) + ww

.= uu + (vv + ww) by RLVECT_1:def 3

.= u + (v + w) ; :: thesis: verum

end;reconsider uu = u, vv = v, ww = w as Element of the carrier of R ;

thus (u + v) + w = (uu + vv) + ww

.= uu + (vv + ww) by RLVECT_1:def 3

.= u + (v + w) ; :: thesis: verum

proof

A25:
ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is right_complementable
let v be Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #); :: according to RLVECT_1:def 4 :: thesis: v + (0. ModuleStr(# the carrier of R, the addF of R,(0. R),lm #)) = v

reconsider vv = v as Element of the carrier of R ;

thus v + (0. ModuleStr(# the carrier of R, the addF of R,(0. R),lm #)) = vv + (0. R)

.= v by RLVECT_1:def 4 ; :: thesis: verum

end;reconsider vv = v as Element of the carrier of R ;

thus v + (0. ModuleStr(# the carrier of R, the addF of R,(0. R),lm #)) = vv + (0. R)

.= v by RLVECT_1:def 4 ; :: thesis: verum

proof

ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) is Abelian
let v be Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #); :: according to ALGSTR_0:def 16 :: thesis: v is right_complementable

reconsider vv = v as Element of the carrier of R ;

consider ww being Element of the carrier of R such that

A26: vv + ww = 0. R by ALGSTR_0:def 11;

reconsider w = ww as Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) ;

v + w = 0. ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) by A26;

hence ex w being Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) st v + w = 0. ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) ; :: according to ALGSTR_0:def 11 :: thesis: verum

end;reconsider vv = v as Element of the carrier of R ;

consider ww being Element of the carrier of R such that

A26: vv + ww = 0. R by ALGSTR_0:def 11;

reconsider w = ww as Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) ;

v + w = 0. ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) by A26;

hence ex w being Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) st v + w = 0. ModuleStr(# the carrier of R, the addF of R,(0. R),lm #) ; :: according to ALGSTR_0:def 11 :: thesis: verum

proof

hence
ex b
let v, w be Element of the carrier of ModuleStr(# the carrier of R, the addF of R,(0. R),lm #); :: according to RLVECT_1:def 2 :: thesis: v + w = w + v

reconsider vv = v, ww = w as Element of the carrier of R ;

thus v + w = ww + vv by RLVECT_1:2

.= w + v ; :: thesis: verum

end;reconsider vv = v, ww = w as Element of the carrier of R ;

thus v + w = ww + vv by RLVECT_1:2

.= w + v ; :: thesis: verum

( addLoopStr(# the carrier of b