Theorem ercpbl 16209

Description: Translate the function compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
ercpbl.r	|- ( ph -> .~ Er V )
ercpbl.v	|- ( ph -> V e. _V )
ercpbl.f	|- F = ( x e. V |-> [ x ] .~ )
ercpbl.c	|- ( ( ph /\ ( a e. V /\ b e. V ) ) -> ( a .+ b ) e. V )
ercpbl.e	|- ( ph -> ( ( A .~ C /\ B .~ D ) -> ( A .+ B ) .~ ( C .+ D ) ) )

Assertion

Ref	Expression
ercpbl	|- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( F ` ( A .+ B ) ) = ( F ` ( C .+ D ) ) ) )

Distinct variable groups:   x, .~   a, b, x, A   B, b, x   x, C   x, D   V, a, b, x   .+ , a, b, x   ph, a, b, x

Allowed substitution hints:   B( a)   C( a, b)   D( a, b)   .~ ( a, b)   F( x, a, b)

Proof of Theorem ercpbl

Step	Hyp	Ref	Expression
1		ercpbl.e	. . 3 |- ( ph -> ( ( A .~ C /\ B .~ D ) -> ( A .+ B ) .~ ( C .+ D ) ) )
2	1	3ad2ant1 1082	. 2 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( A .~ C /\ B .~ D ) -> ( A .+ B ) .~ ( C .+ D ) ) )
3		ercpbl.r	. . . . 5 |- ( ph -> .~ Er V )
4	3	3ad2ant1 1082	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> .~ Er V )
5		ercpbl.v	. . . . 5 |- ( ph -> V e. _V )
6	5	3ad2ant1 1082	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> V e. _V )
7		ercpbl.f	. . . 4 |- F = ( x e. V |-> [ x ] .~ )
8		simp2l 1087	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> A e. V )
9	4, 6, 7, 8	ercpbllem 16208	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( F ` A ) = ( F ` C ) <-> A .~ C ) )
10		simp2r 1088	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> B e. V )
11	4, 6, 7, 10	ercpbllem 16208	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( F ` B ) = ( F ` D ) <-> B .~ D ) )
12	9, 11	anbi12d 747	. 2 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) <-> ( A .~ C /\ B .~ D ) ) )
13		ercpbl.c	. . . . 5 |- ( ( ph /\ ( a e. V /\ b e. V ) ) -> ( a .+ b ) e. V )
14	13	caovclg 6826	. . . 4 |- ( ( ph /\ ( A e. V /\ B e. V ) ) -> ( A .+ B ) e. V )
15	14	3adant3 1081	. . 3 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( A .+ B ) e. V )
16	4, 6, 7, 15	ercpbllem 16208	. 2 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( F ` ( A .+ B ) ) = ( F ` ( C .+ D ) ) <-> ( A .+ B ) .~ ( C .+ D ) ) )
17	2, 12, 16	3imtr4d 283	1 |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( F ` ( A .+ B ) ) = ( F ` ( C .+ D ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Er wer 7739   [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-er 7742  df-ec 7744

This theorem is referenced by:  qusaddvallem  16211  qusaddflem  16212  qusgrp2  17533  qusring2  18620