Theorem erovlem 6521

Description: Lemma for eroprf 6522. (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 ) }

Assertion

Ref	Expression
erovlem	|- ( 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 ) ) ) )

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   R, p, q, r, s, t, u, x, y, z   K, p, q, 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

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)   Z( x, y, z, u, t, s, r, q, p)

Proof of Theorem erovlem

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 108	. . . . . . . 8 |- ( ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) -> ( x = [ p ] R /\ y = [ q ] S ) )
2	1	reximi 2529	. . . . . . 7 |- ( E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) -> E. q e. B ( x = [ p ] R /\ y = [ q ] S ) )
3	2	reximi 2529	. . . . . 6 |- ( 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 ( x = [ p ] R /\ y = [ q ] S ) )
4		eropr.1	. . . . . . . . . 10 |- J = ( A /. R )
5	4	eleq2i 2206	. . . . . . . . 9 |- ( x e. J <-> x e. ( A /. R ) )
6		vex 2689	. . . . . . . . . 10 |- x e. _V
7	6	elqs 6480	. . . . . . . . 9 |- ( x e. ( A /. R ) <-> E. p e. A x = [ p ] R )
8	5, 7	bitri 183	. . . . . . . 8 |- ( x e. J <-> E. p e. A x = [ p ] R )
9		eropr.2	. . . . . . . . . 10 |- K = ( B /. S )
10	9	eleq2i 2206	. . . . . . . . 9 |- ( y e. K <-> y e. ( B /. S ) )
11		vex 2689	. . . . . . . . . 10 |- y e. _V
12	11	elqs 6480	. . . . . . . . 9 |- ( y e. ( B /. S ) <-> E. q e. B y = [ q ] S )
13	10, 12	bitri 183	. . . . . . . 8 |- ( y e. K <-> E. q e. B y = [ q ] S )
14	8, 13	anbi12i 455	. . . . . . 7 |- ( ( x e. J /\ y e. K ) <-> ( E. p e. A x = [ p ] R /\ E. q e. B y = [ q ] S ) )
15		reeanv 2600	. . . . . . 7 |- ( E. p e. A E. q e. B ( x = [ p ] R /\ y = [ q ] S ) <-> ( E. p e. A x = [ p ] R /\ E. q e. B y = [ q ] S ) )
16	14, 15	bitr4i 186	. . . . . 6 |- ( ( x e. J /\ y e. K ) <-> E. p e. A E. q e. B ( x = [ p ] R /\ y = [ q ] S ) )
17	3, 16	sylibr 133	. . . . 5 |- ( E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) -> ( x e. J /\ y e. K ) )
18	17	pm4.71ri 389	. . . 4 |- ( E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> ( ( x e. J /\ y e. K ) /\ E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) )
19		eropr.3	. . . . . . . 8 |- ( ph -> T e. Z )
20		eropr.4	. . . . . . . 8 |- ( ph -> R Er U )
21		eropr.5	. . . . . . . 8 |- ( ph -> S Er V )
22		eropr.6	. . . . . . . 8 |- ( ph -> T Er W )
23		eropr.7	. . . . . . . 8 |- ( ph -> A C_ U )
24		eropr.8	. . . . . . . 8 |- ( ph -> B C_ V )
25		eropr.9	. . . . . . . 8 |- ( ph -> C C_ W )
26		eropr.10	. . . . . . . 8 |- ( ph -> .+ : ( A X. B ) --> C )
27		eropr.11	. . . . . . . 8 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( r R s /\ t S u ) -> ( r .+ t ) T ( s .+ u ) ) )
28	4, 9, 19, 20, 21, 22, 23, 24, 25, 26, 27	eroveu 6520	. . . . . . 7 |- ( ( ph /\ ( x e. J /\ y e. K ) ) -> E! z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
29		iota1 5102	. . . . . . 7 |- ( E! z 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 ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) = z ) )
30	28, 29	syl 14	. . . . . 6 |- ( ( ph /\ ( x e. J /\ y e. K ) ) -> ( 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 ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) = z ) )
31		eqcom 2141	. . . . . 6 |- ( ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) = z <-> z = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) )
32	30, 31	syl6bb 195	. . . . 5 |- ( ( ph /\ ( x e. J /\ y e. K ) ) -> ( E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> z = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) )
33	32	pm5.32da 447	. . . 4 |- ( ph -> ( ( ( x e. J /\ y e. K ) /\ E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) <-> ( ( x e. J /\ y e. K ) /\ z = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) ) )
34	18, 33	syl5bb 191	. . 3 |- ( ph -> ( E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> ( ( x e. J /\ y e. K ) /\ z = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) ) )
35	34	oprabbidv 5825	. 2 |- ( ph -> { <. <. x , y >. , z >. | E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) } = { <. <. x , y >. , z >. | ( ( x e. J /\ y e. K ) /\ z = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) } )
36		eropr.12	. 2 |- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) }
37		df-mpo 5779	. . 3 |- ( 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 ) ) ) = { <. <. x , y >. , w >. | ( ( x e. J /\ y e. K ) /\ w = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) }
38		nfv 1508	. . . 4 |- F/ w ( ( x e. J /\ y e. K ) /\ z = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) )
39		nfv 1508	. . . . 5 |- F/ z ( x e. J /\ y e. K )
40		nfiota1 5090	. . . . . 6 |- F/_ z ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
41	40	nfeq2 2293	. . . . 5 |- F/ z w = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
42	39, 41	nfan 1544	. . . 4 |- F/ z ( ( x e. J /\ y e. K ) /\ w = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) )
43		eqeq1 2146	. . . . 5 |- ( z = w -> ( z = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) <-> w = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) )
44	43	anbi2d 459	. . . 4 |- ( z = w -> ( ( ( x e. J /\ y e. K ) /\ z = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) <-> ( ( x e. J /\ y e. K ) /\ w = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) ) )
45	38, 42, 44	cbvoprab3 5847	. . 3 |- { <. <. x , y >. , z >. | ( ( x e. J /\ y e. K ) /\ z = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) } = { <. <. x , y >. , w >. | ( ( x e. J /\ y e. K ) /\ w = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) }
46	37, 45	eqtr4i 2163	. 2 |- ( 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 ) ) ) = { <. <. x , y >. , z >. | ( ( x e. J /\ y e. K ) /\ z = ( iota z E. p e. A E. q e. B ( ( x = [ p ] R /\ y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) ) ) }
47	35, 36, 46	3eqtr4g 2197	1 |- ( 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 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   E!weu 1999   E.wrex 2417   C_ wss 3071   class class class wbr 3929   X. cxp 4537   iotacio 5086   -->wf 5119  (class class class)co 5774   {coprab 5775   e. cmpo 5776   Er wer 6426   [cec 6427   /.cqs 6428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-er 6429  df-ec 6431  df-qs 6435

This theorem is referenced by:  eroprf  6522