Theorem erov 7844

Description: The value of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
eropr.1	|- J = ( A /. R )
eropr.2	|- K = ( B /. S )
eropr.3	|- ( ph -> T e. Z )
eropr.4	|- ( ph -> R Er U )
eropr.5	|- ( ph -> S Er V )
eropr.6	|- ( ph -> T Er W )
eropr.7	|- ( ph -> A C_ U )
eropr.8	|- ( ph -> B C_ V )
eropr.9	|- ( ph -> C C_ W )
eropr.10	|- ( ph -> .+ : ( A X. B ) --> C )
eropr.11	|- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( r R s /\ t S u ) -> ( r .+ t ) T ( s .+ u ) ) )
eropr.12	|- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) }
eropr.13	|- ( ph -> R e. X )
eropr.14	|- ( ph -> S e. Y )

Assertion

Ref	Expression
erov	|- ( ( ph /\ P e. A /\ Q e. B ) -> ( [ P ] R .+^ [ Q ] S ) = [ ( P .+ Q ) ] T )

Distinct variable groups:   q, p, r, s, t, u, x, y, z, A   B, p, q, r, s, t, u, x, y, z   J, p, q, x, y, z   P, p, q, r, s, t, u, x, y, z   R, p, q, r, s, t, u, x, y, z   K, p, q, x, y, z   Q, p, q, r, s, t, u, x, y, z   S, p, q, r, s, t, u, x, y, z   .+ , p, q, r, s, t, u, x, y, z   ph, p, q, r, s, t, u, x, y, z   T, p, q, r, s, t, u, x, y, z   X, p, q, r, s, t, u, z   Y, p, q, r, s, t, u, z

Allowed substitution hints:   C( x, y, z, u, t, s, r, q, p)   .+^ ( x, y, z, u, t, s, r, q, p)   U( x, y, z, u, t, s, r, q, p)   J( u, t, s, r)   K( u, t, s, r)   V( x, y, z, u, t, s, r, q, p)   W( x, y, z, u, t, s, r, q, p)   X( x, y)   Y( x, y)   Z( x, y, z, u, t, s, r, q, p)

Proof of Theorem erov

Step	Hyp	Ref	Expression
1		eropr.1	. . . . 5 |- J = ( A /. R )
2		eropr.2	. . . . 5 |- K = ( B /. S )
3		eropr.3	. . . . 5 |- ( ph -> T e. Z )
4		eropr.4	. . . . 5 |- ( ph -> R Er U )
5		eropr.5	. . . . 5 |- ( ph -> S Er V )
6		eropr.6	. . . . 5 |- ( ph -> T Er W )
7		eropr.7	. . . . 5 |- ( ph -> A C_ U )
8		eropr.8	. . . . 5 |- ( ph -> B C_ V )
9		eropr.9	. . . . 5 |- ( ph -> C C_ W )
10		eropr.10	. . . . 5 |- ( ph -> .+ : ( A X. B ) --> C )
11		eropr.11	. . . . 5 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( r R s /\ t S u ) -> ( r .+ t ) T ( s .+ u ) ) )
12		eropr.12	. . . . 5 |- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) }
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	erovlem 7843	. . . 4 |- ( ph -> .+^ = ( x e. J , y e. K |-> ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) )
14	13	3ad2ant1 1082	. . 3 |- ( ( ph /\ P e. A /\ Q e. B ) -> .+^ = ( x e. J , y e. K |-> ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) )
15		simprl 794	. . . . . . . 8 |- ( ( ( ph /\ P e. A /\ Q e. B ) /\ ( x = [ P ] R /\ y = [ Q ] S ) ) -> x = [ P ] R )
16	15	eqeq1d 2624	. . . . . . 7 |- ( ( ( ph /\ P e. A /\ Q e. B ) /\ ( x = [ P ] R /\ y = [ Q ] S ) ) -> ( x = [ p ] R <-> [ P ] R = [ p ] R ) )
17		simprr 796	. . . . . . . 8 |- ( ( ( ph /\ P e. A /\ Q e. B ) /\ ( x = [ P ] R /\ y = [ Q ] S ) ) -> y = [ Q ] S )
18	17	eqeq1d 2624	. . . . . . 7 |- ( ( ( ph /\ P e. A /\ Q e. B ) /\ ( x = [ P ] R /\ y = [ Q ] S ) ) -> ( y = [ q ] S <-> [ Q ] S = [ q ] S ) )
19	16, 18	anbi12d 747	. . . . . 6 |- ( ( ( ph /\ P e. A /\ Q e. B ) /\ ( x = [ P ] R /\ y = [ Q ] S ) ) -> ( ( x = [ p ] R /\ y = [ q ] S ) <-> ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) ) )
20	19	anbi1d 741	. . . . 5 |- ( ( ( ph /\ P e. A /\ Q e. B ) /\ ( x = [ P ] R /\ y = [ Q ] S ) ) -> ( ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) )
21	20	2rexbidv 3057	. . . 4 |- ( ( ( ph /\ P e. A /\ Q e. B ) /\ ( x = [ P ] R /\ y = [ Q ] S ) ) -> ( E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) )
22	21	iotabidv 5872	. . 3 |- ( ( ( ph /\ P e. A /\ Q e. B ) /\ ( x = [ P ] R /\ y = [ Q ] S ) ) -> ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) = ( iota z E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) )
23		eropr.13	. . . . 5 |- ( ph -> R e. X )
24		ecelqsg 7802	. . . . . 6 |- ( ( R e. X /\ P e. A ) -> [ P ] R e. ( A /. R ) )
25	24, 1	syl6eleqr 2712	. . . . 5 |- ( ( R e. X /\ P e. A ) -> [ P ] R e. J )
26	23, 25	sylan 488	. . . 4 |- ( ( ph /\ P e. A ) -> [ P ] R e. J )
27	26	3adant3 1081	. . 3 |- ( ( ph /\ P e. A /\ Q e. B ) -> [ P ] R e. J )
28		eropr.14	. . . . 5 |- ( ph -> S e. Y )
29		ecelqsg 7802	. . . . . 6 |- ( ( S e. Y /\ Q e. B ) -> [ Q ] S e. ( B /. S ) )
30	29, 2	syl6eleqr 2712	. . . . 5 |- ( ( S e. Y /\ Q e. B ) -> [ Q ] S e. K )
31	28, 30	sylan 488	. . . 4 |- ( ( ph /\ Q e. B ) -> [ Q ] S e. K )
32	31	3adant2 1080	. . 3 |- ( ( ph /\ P e. A /\ Q e. B ) -> [ Q ] S e. K )
33		iotaex 5868	. . . 4 |- ( iota z E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) e. _V
34	33	a1i 11	. . 3 |- ( ( ph /\ P e. A /\ Q e. B ) -> ( iota z E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) e. _V )
35	14, 22, 27, 32, 34	ovmpt2d 6788	. 2 |- ( ( ph /\ P e. A /\ Q e. B ) -> ( [ P ] R .+^ [ Q ] S ) = ( iota z E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) )
36		eqid 2622	. . . . . . 7 |- [ P ] R = [ P ] R
37		eqid 2622	. . . . . . 7 |- [ Q ] S = [ Q ] S
38	36, 37	pm3.2i 471	. . . . . 6 |- ( [ P ] R = [ P ] R /\ [ Q ] S = [ Q ] S )
39		eqid 2622	. . . . . 6 |- [ ( P .+ Q ) ] T = [ ( P .+ Q ) ] T
40	38, 39	pm3.2i 471	. . . . 5 |- ( ( [ P ] R = [ P ] R /\ [ Q ] S = [ Q ] S ) /\ [ ( P .+ Q ) ] T = [ ( P .+ Q ) ] T )
41		eceq1 7782	. . . . . . . . 9 |- ( p = P -> [ p ] R = [ P ] R )
42	41	eqeq2d 2632	. . . . . . . 8 |- ( p = P -> ( [ P ] R = [ p ] R <-> [ P ] R = [ P ] R ) )
43	42	anbi1d 741	. . . . . . 7 |- ( p = P -> ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) <-> ( [ P ] R = [ P ] R /\ [ Q ] S = [ q ] S ) ) )
44		oveq1 6657	. . . . . . . . 9 |- ( p = P -> ( p .+ q ) = ( P .+ q ) )
45	44	eceq1d 7783	. . . . . . . 8 |- ( p = P -> [ ( p .+ q ) ] T = [ ( P .+ q ) ] T )
46	45	eqeq2d 2632	. . . . . . 7 |- ( p = P -> ( [ ( P .+ Q ) ] T = [ ( p .+ q ) ] T <-> [ ( P .+ Q ) ] T = [ ( P .+ q ) ] T ) )
47	43, 46	anbi12d 747	. . . . . 6 |- ( p = P -> ( ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ [ ( P .+ Q ) ] T = [ ( p .+ q ) ] T ) <-> ( ( [ P ] R = [ P ] R /\ [ Q ] S = [ q ] S ) /\ [ ( P .+ Q ) ] T = [ ( P .+ q ) ] T ) ) )
48		eceq1 7782	. . . . . . . . 9 |- ( q = Q -> [ q ] S = [ Q ] S )
49	48	eqeq2d 2632	. . . . . . . 8 |- ( q = Q -> ( [ Q ] S = [ q ] S <-> [ Q ] S = [ Q ] S ) )
50	49	anbi2d 740	. . . . . . 7 |- ( q = Q -> ( ( [ P ] R = [ P ] R /\ [ Q ] S = [ q ] S ) <-> ( [ P ] R = [ P ] R /\ [ Q ] S = [ Q ] S ) ) )
51		oveq2 6658	. . . . . . . . 9 |- ( q = Q -> ( P .+ q ) = ( P .+ Q ) )
52	51	eceq1d 7783	. . . . . . . 8 |- ( q = Q -> [ ( P .+ q ) ] T = [ ( P .+ Q ) ] T )
53	52	eqeq2d 2632	. . . . . . 7 |- ( q = Q -> ( [ ( P .+ Q ) ] T = [ ( P .+ q ) ] T <-> [ ( P .+ Q ) ] T = [ ( P .+ Q ) ] T ) )
54	50, 53	anbi12d 747	. . . . . 6 |- ( q = Q -> ( ( ( [ P ] R = [ P ] R /\ [ Q ] S = [ q ] S ) /\ [ ( P .+ Q ) ] T = [ ( P .+ q ) ] T ) <-> ( ( [ P ] R = [ P ] R /\ [ Q ] S = [ Q ] S ) /\ [ ( P .+ Q ) ] T = [ ( P .+ Q ) ] T ) ) )
55	47, 54	rspc2ev 3324	. . . . 5 |- ( ( P e. A /\ Q e. B /\ ( ( [ P ] R = [ P ] R /\ [ Q ] S = [ Q ] S ) /\ [ ( P .+ Q ) ] T = [ ( P .+ Q ) ] T ) ) -> E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ [ ( P .+ Q ) ] T = [ ( p .+ q ) ] T ) )
56	40, 55	mp3an3 1413	. . . 4 |- ( ( P e. A /\ Q e. B ) -> E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ [ ( P .+ Q ) ] T = [ ( p .+ q ) ] T ) )
57	56	3adant1 1079	. . 3 |- ( ( ph /\ P e. A /\ Q e. B ) -> E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ [ ( P .+ Q ) ] T = [ ( p .+ q ) ] T ) )
58		ecexg 7746	. . . . . 6 |- ( T e. Z -> [ ( P .+ Q ) ] T e. _V )
59	3, 58	syl 17	. . . . 5 |- ( ph -> [ ( P .+ Q ) ] T e. _V )
60	59	3ad2ant1 1082	. . . 4 |- ( ( ph /\ P e. A /\ Q e. B ) -> [ ( P .+ Q ) ] T e. _V )
61		simp1 1061	. . . . 5 |- ( ( ph /\ P e. A /\ Q e. B ) -> ph )
62	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	eroveu 7842	. . . . 5 |- ( ( ph /\ ( [ P ] R e. J /\ [ Q ] S e. K ) ) -> E! z E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
63	61, 27, 32, 62	syl12anc 1324	. . . 4 |- ( ( ph /\ P e. A /\ Q e. B ) -> E! z E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
64		simpr 477	. . . . . . 7 |- ( ( ( ph /\ P e. A /\ Q e. B ) /\ z = [ ( P .+ Q ) ] T ) -> z = [ ( P .+ Q ) ] T )
65	64	eqeq1d 2624	. . . . . 6 |- ( ( ( ph /\ P e. A /\ Q e. B ) /\ z = [ ( P .+ Q ) ] T ) -> ( z = [ ( p .+ q ) ] T <-> [ ( P .+ Q ) ] T = [ ( p .+ q ) ] T ) )
66	65	anbi2d 740	. . . . 5 |- ( ( ( ph /\ P e. A /\ Q e. B ) /\ z = [ ( P .+ Q ) ] T ) -> ( ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ [ ( P .+ Q ) ] T = [ ( p .+ q ) ] T ) ) )
67	66	2rexbidv 3057	. . . 4 |- ( ( ( ph /\ P e. A /\ Q e. B ) /\ z = [ ( P .+ Q ) ] T ) -> ( E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ [ ( P .+ Q ) ] T = [ ( p .+ q ) ] T ) ) )
68	60, 63, 67	iota2d 5876	. . 3 |- ( ( ph /\ P e. A /\ Q e. B ) -> ( E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ [ ( P .+ Q ) ] T = [ ( p .+ q ) ] T ) <-> ( iota z E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) = [ ( P .+ Q ) ] T ) )
69	57, 68	mpbid 222	. 2 |- ( ( ph /\ P e. A /\ Q e. B ) -> ( iota z E. p e. A E. q e. B ( ( [ P ] R = [ p ] R /\ [ Q ] S = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) = [ ( P .+ Q ) ] T )
70	35, 69	eqtrd 2656	1 |- ( ( ph /\ P e. A /\ Q e. B ) -> ( [ P ] R .+^ [ Q ] S ) = [ ( P .+ Q ) ] T )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E!weu 2470   E.wrex 2913   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   X. cxp 5112   iotacio 5849   -->wf 5884  (class class class)co 6650   {coprab 6651   |-> cmpt2 6652   Er wer 7739   [cec 7740   /.cqs 7741

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-pr 4906  ax-un 6949

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-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-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-ec 7744  df-qs 7748

This theorem is referenced by:  erov2  7846