let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y st a 'eqv' b = I_el Y holds
(All a,PA,G) 'eqv' (All b,PA,G) = I_el Y
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y st a 'eqv' b = I_el Y holds
(All a,PA,G) 'eqv' (All b,PA,G) = I_el Y
let G be Subset of (PARTITIONS Y); :: thesis: for PA being a_partition of Y st a 'eqv' b = I_el Y holds
(All a,PA,G) 'eqv' (All b,PA,G) = I_el Y
let PA be a_partition of Y; :: thesis: ( a 'eqv' b = I_el Y implies (All a,PA,G) 'eqv' (All b,PA,G) = I_el Y )
assume
a 'eqv' b = I_el Y
; :: thesis: (All a,PA,G) 'eqv' (All b,PA,G) = I_el Y
then
( a 'imp' b = I_el Y & b 'imp' a = I_el Y )
by Th10;
then A1:
( ('not' a) 'or' b = I_el Y & ('not' b) 'or' a = I_el Y )
by Th8;
for z being Element of Y holds ((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
proof
let z be
Element of
Y;
:: thesis: ((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
(All a,PA,G) 'eqv' (All b,PA,G) =
((All a,PA,G) 'imp' (All b,PA,G)) '&' ((All b,PA,G) 'imp' (All a,PA,G))
by Th7
.=
(('not' (All a,PA,G)) 'or' (All b,PA,G)) '&' ((All b,PA,G) 'imp' (All a,PA,G))
by Th8
.=
(('not' (All a,PA,G)) 'or' (All b,PA,G)) '&' (('not' (All b,PA,G)) 'or' (All a,PA,G))
by Th8
.=
((('not' (All a,PA,G)) 'or' (All b,PA,G)) '&' ('not' (All b,PA,G))) 'or' ((('not' (All a,PA,G)) 'or' (All b,PA,G)) '&' (All a,PA,G))
by BVFUNC_1:15
.=
((('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) 'or' ((All b,PA,G) '&' ('not' (All b,PA,G)))) 'or' ((('not' (All a,PA,G)) 'or' (All b,PA,G)) '&' (All a,PA,G))
by BVFUNC_1:15
.=
((('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) 'or' ((All b,PA,G) '&' ('not' (All b,PA,G)))) 'or' ((('not' (All a,PA,G)) '&' (All a,PA,G)) 'or' ((All b,PA,G) '&' (All a,PA,G)))
by BVFUNC_1:15
.=
((('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) 'or' (O_el Y)) 'or' ((('not' (All a,PA,G)) '&' (All a,PA,G)) 'or' ((All b,PA,G) '&' (All a,PA,G)))
by Th5
.=
((('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) 'or' (O_el Y)) 'or' ((O_el Y) 'or' ((All b,PA,G) '&' (All a,PA,G)))
by Th5
.=
(('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) 'or' ((O_el Y) 'or' ((All b,PA,G) '&' (All a,PA,G)))
by BVFUNC_1:12
.=
(('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) 'or' ((All b,PA,G) '&' (All a,PA,G))
by BVFUNC_1:12
;
then A2:
((All a,PA,G) 'eqv' (All b,PA,G)) . z =
((('not' (All a,PA,G)) '&' ('not' (All b,PA,G))) . z) 'or' (((All b,PA,G) '&' (All a,PA,G)) . z)
by BVFUNC_1:def 7
.=
((('not' (All a,PA,G)) . z) '&' (('not' (All b,PA,G)) . z)) 'or' (((All b,PA,G) '&' (All a,PA,G)) . z)
by MARGREL1:def 21
.=
((('not' (All a,PA,G)) . z) '&' (('not' (All b,PA,G)) . z)) 'or' (((All b,PA,G) . z) '&' ((All a,PA,G) . z))
by MARGREL1:def 21
.=
(('not' ((All a,PA,G) . z)) '&' (('not' (All b,PA,G)) . z)) 'or' (((All b,PA,G) . z) '&' ((All a,PA,G) . z))
by MARGREL1:def 20
.=
(('not' ((All a,PA,G) . z)) '&' ('not' ((All b,PA,G) . z))) 'or' (((All b,PA,G) . z) '&' ((All a,PA,G) . z))
by MARGREL1:def 20
;
A3:
now assume A4:
( ( for
x being
Element of
Y st
x in EqClass z,
(CompF PA,G) holds
a . x = TRUE ) & ( for
x being
Element of
Y st
x in EqClass z,
(CompF PA,G) holds
b . x = TRUE ) )
;
:: thesis: ((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE then
(B_INF b,(CompF PA,G)) . z = TRUE
by BVFUNC_1:def 19;
hence
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
by A2, A4, BVFUNC_1:def 19;
:: thesis: verum end;
A5:
now assume A6:
( ( for
x being
Element of
Y st
x in EqClass z,
(CompF PA,G) holds
a . x = TRUE ) & ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
b . x = TRUE ) )
;
:: thesis: ((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE then consider x1 being
Element of
Y such that A7:
(
x1 in EqClass z,
(CompF PA,G) &
b . x1 <> TRUE )
;
A8:
b . x1 = FALSE
by A7, XBOOLEAN:def 3;
A9:
a . x1 = TRUE
by A6, A7;
(('not' a) 'or' b) . x1 =
(('not' a) . x1) 'or' (b . x1)
by BVFUNC_1:def 7
.=
FALSE 'or' FALSE
by A8, A9, MARGREL1:def 20
.=
FALSE
;
hence
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
by A1, BVFUNC_1:def 14;
:: thesis: verum end;
A10:
now assume A11:
( ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
a . x = TRUE ) & ( for
x being
Element of
Y st
x in EqClass z,
(CompF PA,G) holds
b . x = TRUE ) )
;
:: thesis: ((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE then consider x1 being
Element of
Y such that A12:
(
x1 in EqClass z,
(CompF PA,G) &
a . x1 <> TRUE )
;
A13:
a . x1 = FALSE
by A12, XBOOLEAN:def 3;
A14:
b . x1 = TRUE
by A11, A12;
(('not' b) 'or' a) . x1 =
(('not' b) . x1) 'or' (a . x1)
by BVFUNC_1:def 7
.=
FALSE 'or' FALSE
by A13, A14, MARGREL1:def 20
.=
FALSE
;
hence
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
by A1, BVFUNC_1:def 14;
:: thesis: verum end;
now assume A15:
( ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
a . x = TRUE ) & ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
b . x = TRUE ) )
;
:: thesis: ((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE then
(B_INF b,(CompF PA,G)) . z = FALSE
by BVFUNC_1:def 19;
hence
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
by A2, A15, BVFUNC_1:def 19;
:: thesis: verum end;
hence
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
by A3, A5, A10;
:: thesis: verum
end;
hence
(All a,PA,G) 'eqv' (All b,PA,G) = I_el Y
by BVFUNC_1:def 14; :: thesis: verum