Theorem eroveu 6592

Description: Lemma for eroprf 6594. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-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 ) ) )

Assertion

Ref	Expression
eroveu	|- ( ( 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 ) )

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

Allowed substitution hints:   C( z, u, t, s, r, q, p)   U( z, u, t, s, r, q, p)   J( u, t, s, r)   K( u, t, s, r)   V( z, u, t, s, r, q, p)   W( z, u, t, s, r, q, p)   Z( z, u, t, s, r, q, p)

Proof of Theorem eroveu

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqsi 6553	. . . . . . . 8 |- ( X e. ( A /. R ) -> E. p e. A X = [ p ] R )
2		eropr.1	. . . . . . . 8 |- J = ( A /. R )
3	1, 2	eleq2s 2261	. . . . . . 7 |- ( X e. J -> E. p e. A X = [ p ] R )
4		elqsi 6553	. . . . . . . 8 |- ( Y e. ( B /. S ) -> E. q e. B Y = [ q ] S )
5		eropr.2	. . . . . . . 8 |- K = ( B /. S )
6	4, 5	eleq2s 2261	. . . . . . 7 |- ( Y e. K -> E. q e. B Y = [ q ] S )
7	3, 6	anim12i 336	. . . . . 6 |- ( ( X e. J /\ Y e. K ) -> ( E. p e. A X = [ p ] R /\ E. q e. B Y = [ q ] S ) )
8	7	adantl 275	. . . . 5 |- ( ( ph /\ ( X e. J /\ Y e. K ) ) -> ( E. p e. A X = [ p ] R /\ E. q e. B Y = [ q ] S ) )
9		reeanv 2635	. . . . 5 |- ( 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 ) )
10	8, 9	sylibr 133	. . . 4 |- ( ( ph /\ ( X e. J /\ Y e. K ) ) -> E. p e. A E. q e. B ( X = [ p ] R /\ Y = [ q ] S ) )
11		eropr.3	. . . . . . . 8 |- ( ph -> T e. Z )
12	11	adantr 274	. . . . . . 7 |- ( ( ph /\ ( X e. J /\ Y e. K ) ) -> T e. Z )
13		ecexg 6505	. . . . . . 7 |- ( T e. Z -> [ ( p .+ q ) ] T e. _V )
14		elisset 2740	. . . . . . 7 |- ( [ ( p .+ q ) ] T e. _V -> E. z z = [ ( p .+ q ) ] T )
15	12, 13, 14	3syl 17	. . . . . 6 |- ( ( ph /\ ( X e. J /\ Y e. K ) ) -> E. z z = [ ( p .+ q ) ] T )
16	15	biantrud 302	. . . . 5 |- ( ( ph /\ ( X e. J /\ Y e. K ) ) -> ( ( X = [ p ] R /\ Y = [ q ] S ) <-> ( ( X = [ p ] R /\ Y = [ q ] S ) /\ E. z z = [ ( p .+ q ) ] T ) ) )
17	16	2rexbidv 2491	. . . 4 |- ( ( ph /\ ( X e. J /\ Y e. K ) ) -> ( E. p e. A E. q e. B ( X = [ p ] R /\ Y = [ q ] S ) <-> E. p e. A E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ E. z z = [ ( p .+ q ) ] T ) ) )
18	10, 17	mpbid 146	. . 3 |- ( ( ph /\ ( X e. J /\ Y e. K ) ) -> E. p e. A E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ E. z z = [ ( p .+ q ) ] T ) )
19		19.42v 1894	. . . . . . . 8 |- ( E. z ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> ( ( X = [ p ] R /\ Y = [ q ] S ) /\ E. z z = [ ( p .+ q ) ] T ) )
20	19	bicomi 131	. . . . . . 7 |- ( ( ( X = [ p ] R /\ Y = [ q ] S ) /\ E. z z = [ ( p .+ q ) ] T ) <-> E. z ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
21	20	rexbii 2473	. . . . . 6 |- ( E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ E. z z = [ ( p .+ q ) ] T ) <-> E. q e. B E. z ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
22		rexcom4 2749	. . . . . 6 |- ( E. q e. B E. z ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> E. z E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
23	21, 22	bitri 183	. . . . 5 |- ( E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ E. z z = [ ( p .+ q ) ] T ) <-> E. z E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
24	23	rexbii 2473	. . . 4 |- ( E. p e. A E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ E. z z = [ ( p .+ q ) ] T ) <-> E. p e. A E. z E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
25		rexcom4 2749	. . . 4 |- ( E. p e. A E. z E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> E. z E. p e. A E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
26	24, 25	bitri 183	. . 3 |- ( E. p e. A E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ E. z z = [ ( p .+ q ) ] T ) <-> E. z E. p e. A E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) )
27	18, 26	sylib 121	. 2 |- ( ( 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 ) )
28		reeanv 2635	. . . . . 6 |- ( E. r e. A E. s e. A ( E. t e. B ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) /\ E. u e. B ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) <-> ( E. r e. A E. t e. B ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) /\ E. s e. A E. u e. B ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) )
29		eceq1 6536	. . . . . . . . . . 11 |- ( p = r -> [ p ] R = [ r ] R )
30	29	eqeq2d 2177	. . . . . . . . . 10 |- ( p = r -> ( X = [ p ] R <-> X = [ r ] R ) )
31	30	anbi1d 461	. . . . . . . . 9 |- ( p = r -> ( ( X = [ p ] R /\ Y = [ q ] S ) <-> ( X = [ r ] R /\ Y = [ q ] S ) ) )
32		oveq1 5849	. . . . . . . . . . 11 |- ( p = r -> ( p .+ q ) = ( r .+ q ) )
33	32	eceq1d 6537	. . . . . . . . . 10 |- ( p = r -> [ ( p .+ q ) ] T = [ ( r .+ q ) ] T )
34	33	eqeq2d 2177	. . . . . . . . 9 |- ( p = r -> ( z = [ ( p .+ q ) ] T <-> z = [ ( r .+ q ) ] T ) )
35	31, 34	anbi12d 465	. . . . . . . 8 |- ( p = r -> ( ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> ( ( X = [ r ] R /\ Y = [ q ] S ) /\ z = [ ( r .+ q ) ] T ) ) )
36		eceq1 6536	. . . . . . . . . . 11 |- ( q = t -> [ q ] S = [ t ] S )
37	36	eqeq2d 2177	. . . . . . . . . 10 |- ( q = t -> ( Y = [ q ] S <-> Y = [ t ] S ) )
38	37	anbi2d 460	. . . . . . . . 9 |- ( q = t -> ( ( X = [ r ] R /\ Y = [ q ] S ) <-> ( X = [ r ] R /\ Y = [ t ] S ) ) )
39		oveq2 5850	. . . . . . . . . . 11 |- ( q = t -> ( r .+ q ) = ( r .+ t ) )
40	39	eceq1d 6537	. . . . . . . . . 10 |- ( q = t -> [ ( r .+ q ) ] T = [ ( r .+ t ) ] T )
41	40	eqeq2d 2177	. . . . . . . . 9 |- ( q = t -> ( z = [ ( r .+ q ) ] T <-> z = [ ( r .+ t ) ] T ) )
42	38, 41	anbi12d 465	. . . . . . . 8 |- ( q = t -> ( ( ( X = [ r ] R /\ Y = [ q ] S ) /\ z = [ ( r .+ q ) ] T ) <-> ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) ) )
43	35, 42	cbvrex2v 2706	. . . . . . 7 |- ( E. p e. A E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> E. r e. A E. t e. B ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) )
44		eceq1 6536	. . . . . . . . . . 11 |- ( p = s -> [ p ] R = [ s ] R )
45	44	eqeq2d 2177	. . . . . . . . . 10 |- ( p = s -> ( X = [ p ] R <-> X = [ s ] R ) )
46	45	anbi1d 461	. . . . . . . . 9 |- ( p = s -> ( ( X = [ p ] R /\ Y = [ q ] S ) <-> ( X = [ s ] R /\ Y = [ q ] S ) ) )
47		oveq1 5849	. . . . . . . . . . 11 |- ( p = s -> ( p .+ q ) = ( s .+ q ) )
48	47	eceq1d 6537	. . . . . . . . . 10 |- ( p = s -> [ ( p .+ q ) ] T = [ ( s .+ q ) ] T )
49	48	eqeq2d 2177	. . . . . . . . 9 |- ( p = s -> ( w = [ ( p .+ q ) ] T <-> w = [ ( s .+ q ) ] T ) )
50	46, 49	anbi12d 465	. . . . . . . 8 |- ( p = s -> ( ( ( X = [ p ] R /\ Y = [ q ] S ) /\ w = [ ( p .+ q ) ] T ) <-> ( ( X = [ s ] R /\ Y = [ q ] S ) /\ w = [ ( s .+ q ) ] T ) ) )
51		eceq1 6536	. . . . . . . . . . 11 |- ( q = u -> [ q ] S = [ u ] S )
52	51	eqeq2d 2177	. . . . . . . . . 10 |- ( q = u -> ( Y = [ q ] S <-> Y = [ u ] S ) )
53	52	anbi2d 460	. . . . . . . . 9 |- ( q = u -> ( ( X = [ s ] R /\ Y = [ q ] S ) <-> ( X = [ s ] R /\ Y = [ u ] S ) ) )
54		oveq2 5850	. . . . . . . . . . 11 |- ( q = u -> ( s .+ q ) = ( s .+ u ) )
55	54	eceq1d 6537	. . . . . . . . . 10 |- ( q = u -> [ ( s .+ q ) ] T = [ ( s .+ u ) ] T )
56	55	eqeq2d 2177	. . . . . . . . 9 |- ( q = u -> ( w = [ ( s .+ q ) ] T <-> w = [ ( s .+ u ) ] T ) )
57	53, 56	anbi12d 465	. . . . . . . 8 |- ( q = u -> ( ( ( X = [ s ] R /\ Y = [ q ] S ) /\ w = [ ( s .+ q ) ] T ) <-> ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) )
58	50, 57	cbvrex2v 2706	. . . . . . 7 |- ( E. p e. A E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ w = [ ( p .+ q ) ] T ) <-> E. s e. A E. u e. B ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) )
59	43, 58	anbi12i 456	. . . . . 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 ) /\ w = [ ( p .+ q ) ] T ) ) <-> ( E. r e. A E. t e. B ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) /\ E. s e. A E. u e. B ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) )
60	28, 59	bitr4i 186	. . . . 5 |- ( E. r e. A E. s e. A ( E. t e. B ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) /\ E. u e. B ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) <-> ( 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 ) /\ w = [ ( p .+ q ) ] T ) ) )
61		reeanv 2635	. . . . . . 7 |- ( E. t e. B E. u e. B ( ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) /\ ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) <-> ( E. t e. B ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) /\ E. u e. B ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) )
62		eropr.11	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( r R s /\ t S u ) -> ( r .+ t ) T ( s .+ u ) ) )
63		eropr.4	. . . . . . . . . . . . . . . . 17 |- ( ph -> R Er U )
64	63	adantr 274	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> R Er U )
65		eropr.7	. . . . . . . . . . . . . . . . . 18 |- ( ph -> A C_ U )
66	65	adantr 274	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> A C_ U )
67		simprll 527	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> r e. A )
68	66, 67	sseldd 3143	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> r e. U )
69	64, 68	erth 6545	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( r R s <-> [ r ] R = [ s ] R ) )
70		eropr.5	. . . . . . . . . . . . . . . . 17 |- ( ph -> S Er V )
71	70	adantr 274	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> S Er V )
72		eropr.8	. . . . . . . . . . . . . . . . . 18 |- ( ph -> B C_ V )
73	72	adantr 274	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> B C_ V )
74		simprrl 529	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> t e. B )
75	73, 74	sseldd 3143	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> t e. V )
76	71, 75	erth 6545	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( t S u <-> [ t ] S = [ u ] S ) )
77	69, 76	anbi12d 465	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( r R s /\ t S u ) <-> ( [ r ] R = [ s ] R /\ [ t ] S = [ u ] S ) ) )
78		eropr.6	. . . . . . . . . . . . . . . 16 |- ( ph -> T Er W )
79	78	adantr 274	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> T Er W )
80		eropr.9	. . . . . . . . . . . . . . . . 17 |- ( ph -> C C_ W )
81	80	adantr 274	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> C C_ W )
82		eropr.10	. . . . . . . . . . . . . . . . . 18 |- ( ph -> .+ : ( A X. B ) --> C )
83	82	adantr 274	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> .+ : ( A X. B ) --> C )
84	83, 67, 74	fovrnd 5986	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( r .+ t ) e. C )
85	81, 84	sseldd 3143	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( r .+ t ) e. W )
86	79, 85	erth 6545	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( r .+ t ) T ( s .+ u ) <-> [ ( r .+ t ) ] T = [ ( s .+ u ) ] T ) )
87	62, 77, 86	3imtr3d 201	. . . . . . . . . . . . 13 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( [ r ] R = [ s ] R /\ [ t ] S = [ u ] S ) -> [ ( r .+ t ) ] T = [ ( s .+ u ) ] T ) )
88		eqeq2 2175	. . . . . . . . . . . . . 14 |- ( w = [ ( s .+ u ) ] T -> ( [ ( r .+ t ) ] T = w <-> [ ( r .+ t ) ] T = [ ( s .+ u ) ] T ) )
89	88	biimprcd 159	. . . . . . . . . . . . 13 |- ( [ ( r .+ t ) ] T = [ ( s .+ u ) ] T -> ( w = [ ( s .+ u ) ] T -> [ ( r .+ t ) ] T = w ) )
90	87, 89	syl6 33	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( [ r ] R = [ s ] R /\ [ t ] S = [ u ] S ) -> ( w = [ ( s .+ u ) ] T -> [ ( r .+ t ) ] T = w ) ) )
91	90	impd 252	. . . . . . . . . . 11 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( ( [ r ] R = [ s ] R /\ [ t ] S = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) -> [ ( r .+ t ) ] T = w ) )
92		eqeq1 2172	. . . . . . . . . . . . . . 15 |- ( X = [ r ] R -> ( X = [ s ] R <-> [ r ] R = [ s ] R ) )
93		eqeq1 2172	. . . . . . . . . . . . . . 15 |- ( Y = [ t ] S -> ( Y = [ u ] S <-> [ t ] S = [ u ] S ) )
94	92, 93	bi2anan9 596	. . . . . . . . . . . . . 14 |- ( ( X = [ r ] R /\ Y = [ t ] S ) -> ( ( X = [ s ] R /\ Y = [ u ] S ) <-> ( [ r ] R = [ s ] R /\ [ t ] S = [ u ] S ) ) )
95	94	anbi1d 461	. . . . . . . . . . . . 13 |- ( ( X = [ r ] R /\ Y = [ t ] S ) -> ( ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) <-> ( ( [ r ] R = [ s ] R /\ [ t ] S = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) )
96	95	adantr 274	. . . . . . . . . . . 12 |- ( ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) -> ( ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) <-> ( ( [ r ] R = [ s ] R /\ [ t ] S = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) )
97		eqeq1 2172	. . . . . . . . . . . . 13 |- ( z = [ ( r .+ t ) ] T -> ( z = w <-> [ ( r .+ t ) ] T = w ) )
98	97	adantl 275	. . . . . . . . . . . 12 |- ( ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) -> ( z = w <-> [ ( r .+ t ) ] T = w ) )
99	96, 98	imbi12d 233	. . . . . . . . . . 11 |- ( ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) -> ( ( ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) -> z = w ) <-> ( ( ( [ r ] R = [ s ] R /\ [ t ] S = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) -> [ ( r .+ t ) ] T = w ) ) )
100	91, 99	syl5ibrcom 156	. . . . . . . . . 10 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) -> ( ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) -> z = w ) ) )
101	100	impd 252	. . . . . . . . 9 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. B /\ u e. B ) ) ) -> ( ( ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) /\ ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) -> z = w ) )
102	101	anassrs 398	. . . . . . . 8 |- ( ( ( ph /\ ( r e. A /\ s e. A ) ) /\ ( t e. B /\ u e. B ) ) -> ( ( ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) /\ ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) -> z = w ) )
103	102	rexlimdvva 2591	. . . . . . 7 |- ( ( ph /\ ( r e. A /\ s e. A ) ) -> ( E. t e. B E. u e. B ( ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) /\ ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) -> z = w ) )
104	61, 103	syl5bir 152	. . . . . 6 |- ( ( ph /\ ( r e. A /\ s e. A ) ) -> ( ( E. t e. B ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) /\ E. u e. B ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) -> z = w ) )
105	104	rexlimdvva 2591	. . . . 5 |- ( ph -> ( E. r e. A E. s e. A ( E. t e. B ( ( X = [ r ] R /\ Y = [ t ] S ) /\ z = [ ( r .+ t ) ] T ) /\ E. u e. B ( ( X = [ s ] R /\ Y = [ u ] S ) /\ w = [ ( s .+ u ) ] T ) ) -> z = w ) )
106	60, 105	syl5bir 152	. . . 4 |- ( ph -> ( ( 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 ) /\ w = [ ( p .+ q ) ] T ) ) -> z = w ) )
107	106	adantr 274	. . 3 |- ( ( 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 ) /\ E. p e. A E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ w = [ ( p .+ q ) ] T ) ) -> z = w ) )
108	107	alrimivv 1863	. 2 |- ( ( ph /\ ( X e. J /\ Y e. K ) ) -> A. z A. w ( ( 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 ) /\ w = [ ( p .+ q ) ] T ) ) -> z = w ) )
109		eqeq1 2172	. . . . 5 |- ( z = w -> ( z = [ ( p .+ q ) ] T <-> w = [ ( p .+ q ) ] T ) )
110	109	anbi2d 460	. . . 4 |- ( z = w -> ( ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> ( ( X = [ p ] R /\ Y = [ q ] S ) /\ w = [ ( p .+ q ) ] T ) ) )
111	110	2rexbidv 2491	. . 3 |- ( z = w -> ( 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 ) /\ w = [ ( p .+ q ) ] T ) ) )
112	111	eu4 2076	. 2 |- ( E! z E. p e. A E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) <-> ( E. z E. p e. A E. q e. B ( ( X = [ p ] R /\ Y = [ q ] S ) /\ z = [ ( p .+ q ) ] T ) /\ A. z A. w ( ( 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 ) /\ w = [ ( p .+ q ) ] T ) ) -> z = w ) ) )
113	27, 108, 112	sylanbrc 414	1 |- ( ( 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 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   = wceq 1343   E.wex 1480   E!weu 2014   e. wcel 2136   E.wrex 2445   _Vcvv 2726   C_ wss 3116   class class class wbr 3982   X. cxp 4602   -->wf 5184  (class class class)co 5842   Er wer 6498   [cec 6499   /.cqs 6500

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-er 6501  df-ec 6503  df-qs 6507

This theorem is referenced by:  erovlem  6593  eroprf  6594