let K be Ring; :: thesis: for V being LeftMod of K holds opp V is strict RightMod of opp K
let V be LeftMod of K; :: thesis: opp V is strict RightMod of opp K
set R = opp K;
reconsider W = opp V as non empty RightModStr over opp K ;
A1: addLoopStr(# the carrier of (opp V), the addF of (opp V), the ZeroF of (opp V) #) = addLoopStr(# the carrier of V, the addF of V, the ZeroF of V #) by Th7;
then A2: for a, b being Element of (opp V)
for x, y being Element of V st x = a & b = y holds
a + b = x + y ;
A3: ( opp V is Abelian & opp V is add-associative & opp V is right_zeroed & opp V is right_complementable )
proof
thus opp V is Abelian :: thesis:
proof
let a, b be Element of (opp V); :: according to RLVECT_1:def 2 :: thesis: a + b = b + a
reconsider x = a, y = b as Element of V by Th7;
thus a + b = y + x by A2
.= b + a by A1 ; :: thesis: verum
end;
hereby :: according to RLVECT_1:def 3 :: thesis:
let a, b, c be Element of (opp V); :: thesis: (a + b) + c = a + (b + c)
reconsider x = a, y = b, z = c as Element of V by Th7;
thus (a + b) + c = (x + y) + z by A1
.= x + (y + z) by RLVECT_1:def 3
.= a + (b + c) by A1 ; :: thesis: verum
end;
hereby :: according to RLVECT_1:def 4 :: thesis:
let a be Element of (opp V); :: thesis: a + (0. (opp V)) = a
reconsider x = a as Element of V by Th7;
thus a + (0. (opp V)) = x + (0. V) by A1
.= a by RLVECT_1:4 ; :: thesis: verum
end;
let a be Element of (opp V); :: according to ALGSTR_0:def 16 :: thesis:
reconsider x = a as Element of V by Th7;
consider b being Element of V such that
A4: x + b = 0. V by ALGSTR_0:def 11;
reconsider b9 = b as Element of (opp V) by Th7;
take b9 ; :: according to ALGSTR_0:def 11 :: thesis: a + b9 = 0. (opp V)
thus a + b9 = 0. (opp V) by A1, A4; :: thesis: verum
end;
now :: thesis: for x, y being Scalar of (opp K)
for v, w being Vector of W holds
( (v + w) * x = (v * x) + (w * x) & v * (x + y) = (v * x) + (v * y) & v * (y * x) = (v * y) * x & v * (1_ (opp K)) = v )
let x, y be Scalar of (opp K); :: thesis: for v, w being Vector of W holds
( (v + w) * x = (v * x) + (w * x) & v * (x + y) = (v * x) + (v * y) & v * (y * x) = (v * y) * x & v * (1_ (opp K)) = v )

let v, w be Vector of W; :: thesis: ( (v + w) * x = (v * x) + (w * x) & v * (x + y) = (v * x) + (v * y) & v * (y * x) = (v * y) * x & v * (1_ (opp K)) = v )
reconsider p = v, q = w as Vector of V by Th7;
reconsider a = x, b = y as Scalar of K ;
A5: b * p = v * y by Th12;
A6: a * q = w * x by Th12;
A7: a * p = v * x by Th12;
v + w = p + q by Th13;
hence (v + w) * x = a * (p + q) by Th12
.= (a * p) + (a * q) by VECTSP_1:def 14
.= (v * x) + (w * x) by A7, A6, Th13 ;
:: thesis: ( v * (x + y) = (v * x) + (v * y) & v * (y * x) = (v * y) * x & v * (1_ (opp K)) = v )
thus v * (x + y) = (a + b) * p by Th12
.= (a * p) + (b * p) by VECTSP_1:def 15
.= (v * x) + (v * y) by A5, A7, Th13 ; :: thesis: ( v * (y * x) = (v * y) * x & v * (1_ (opp K)) = v )
thus v * (y * x) = (a * b) * p by
.= a * (b * p) by VECTSP_1:def 16
.= (v * y) * x by ; :: thesis: v * (1_ (opp K)) = v
thus v * (1_ (opp K)) = (1_ K) * p by Th12
.= v by VECTSP_1:def 17 ; :: thesis: verum
end;
hence opp V is strict RightMod of opp K by ; :: thesis: verum