Theorem caucvgprprlemnkeqj 7440

Description: Lemma for caucvgprpr 7462. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)

Hypotheses

Ref	Expression
caucvgprpr.f	|- ( ph -> F : N. --> P. )
caucvgprpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
caucvgprprlemnkj.k	|- ( ph -> K e. N. )
caucvgprprlemnkj.j	|- ( ph -> J e. N. )
caucvgprprlemnkj.s	|- ( ph -> S e. Q. )

Assertion

Ref	Expression
caucvgprprlemnkeqj	|- ( ( ph /\ K = J ) -> -. ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) )

Distinct variable groups:   k, F, n   J, p, q   K, p, q   S, p, q

Allowed substitution hints:   ph( u, k, n, q, p, l)   S( u, k, n, l)   F( u, q, p, l)   J( u, k, n, l)   K( u, k, n, l)

Proof of Theorem caucvgprprlemnkeqj

Step	Hyp	Ref	Expression
1		ltsopr 7346	. . . 4 |- <P Or P.
2		ltrelpr 7255	. . . 4 |- <P C_ ( P. X. P. )
3	1, 2	son2lpi 4891	. . 3 |- -. ( ( F ` J ) <P <. { p | p <Q S } , { q | S <Q q } >. /\ <. { p | p <Q S } , { q | S <Q q } >. <P ( F ` J ) )
4		caucvgprpr.f	. . . . . . . . 9 |- ( ph -> F : N. --> P. )
5		caucvgprprlemnkj.j	. . . . . . . . 9 |- ( ph -> J e. N. )
6	4, 5	ffvelrnd 5508	. . . . . . . 8 |- ( ph -> ( F ` J ) e. P. )
7	6	ad2antrr 477	. . . . . . 7 |- ( ( ( ph /\ K = J ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> ( F ` J ) e. P. )
8	5	adantr 272	. . . . . . . . . . 11 |- ( ( ph /\ K = J ) -> J e. N. )
9		nnnq 7172	. . . . . . . . . . 11 |- ( J e. N. -> [ <. J , 1o >. ] ~Q e. Q. )
10	8, 9	syl 14	. . . . . . . . . 10 |- ( ( ph /\ K = J ) -> [ <. J , 1o >. ] ~Q e. Q. )
11		recclnq 7142	. . . . . . . . . 10 |- ( [ <. J , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. )
12	10, 11	syl 14	. . . . . . . . 9 |- ( ( ph /\ K = J ) -> ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. )
13		nqprlu 7297	. . . . . . . . 9 |- ( ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. -> <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. e. P. )
14	12, 13	syl 14	. . . . . . . 8 |- ( ( ph /\ K = J ) -> <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. e. P. )
15	14	adantr 272	. . . . . . 7 |- ( ( ( ph /\ K = J ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. e. P. )
16		ltaddpr 7347	. . . . . . 7 |- ( ( ( F ` J ) e. P. /\ <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( F ` J ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
17	7, 15, 16	syl2anc 406	. . . . . 6 |- ( ( ( ph /\ K = J ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> ( F ` J ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
18		simprr 504	. . . . . 6 |- ( ( ( ph /\ K = J ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. )
19	1, 2	sotri 4890	. . . . . 6 |- ( ( ( F ` J ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) -> ( F ` J ) <P <. { p | p <Q S } , { q | S <Q q } >. )
20	17, 18, 19	syl2anc 406	. . . . 5 |- ( ( ( ph /\ K = J ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> ( F ` J ) <P <. { p | p <Q S } , { q | S <Q q } >. )
21		caucvgprprlemnkj.s	. . . . . . . . . 10 |- ( ph -> S e. Q. )
22	21	adantr 272	. . . . . . . . 9 |- ( ( ph /\ K = J ) -> S e. Q. )
23		nqprlu 7297	. . . . . . . . 9 |- ( S e. Q. -> <. { p | p <Q S } , { q | S <Q q } >. e. P. )
24	22, 23	syl 14	. . . . . . . 8 |- ( ( ph /\ K = J ) -> <. { p | p <Q S } , { q | S <Q q } >. e. P. )
25		ltaddpr 7347	. . . . . . . 8 |- ( ( <. { p | p <Q S } , { q | S <Q q } >. e. P. /\ <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. e. P. ) -> <. { p | p <Q S } , { q | S <Q q } >. <P ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
26	24, 14, 25	syl2anc 406	. . . . . . 7 |- ( ( ph /\ K = J ) -> <. { p | p <Q S } , { q | S <Q q } >. <P ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
27	26	adantr 272	. . . . . 6 |- ( ( ( ph /\ K = J ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> <. { p | p <Q S } , { q | S <Q q } >. <P ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
28		simprl 503	. . . . . . 7 |- ( ( ( ph /\ K = J ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) )
29		addnqpr 7311	. . . . . . . . . 10 |- ( ( S e. Q. /\ ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. ) -> <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. = ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
30	22, 12, 29	syl2anc 406	. . . . . . . . 9 |- ( ( ph /\ K = J ) -> <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. = ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
31	30	breq1d 3903	. . . . . . . 8 |- ( ( ph /\ K = J ) -> ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) <-> ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` J ) ) )
32	31	adantr 272	. . . . . . 7 |- ( ( ( ph /\ K = J ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) <-> ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` J ) ) )
33	28, 32	mpbid 146	. . . . . 6 |- ( ( ( ph /\ K = J ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` J ) )
34	1, 2	sotri 4890	. . . . . 6 |- ( ( <. { p | p <Q S } , { q | S <Q q } >. <P ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) /\ ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` J ) ) -> <. { p | p <Q S } , { q | S <Q q } >. <P ( F ` J ) )
35	27, 33, 34	syl2anc 406	. . . . 5 |- ( ( ( ph /\ K = J ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> <. { p | p <Q S } , { q | S <Q q } >. <P ( F ` J ) )
36	20, 35	jca 302	. . . 4 |- ( ( ( ph /\ K = J ) /\ ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) -> ( ( F ` J ) <P <. { p | p <Q S } , { q | S <Q q } >. /\ <. { p | p <Q S } , { q | S <Q q } >. <P ( F ` J ) ) )
37	36	ex 114	. . 3 |- ( ( ph /\ K = J ) -> ( ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) -> ( ( F ` J ) <P <. { p | p <Q S } , { q | S <Q q } >. /\ <. { p | p <Q S } , { q | S <Q q } >. <P ( F ` J ) ) ) )
38	3, 37	mtoi 636	. 2 |- ( ( ph /\ K = J ) -> -. ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) )
39		opeq1 3669	. . . . . . . . . . 11 |- ( K = J -> <. K , 1o >. = <. J , 1o >. )
40	39	eceq1d 6417	. . . . . . . . . 10 |- ( K = J -> [ <. K , 1o >. ] ~Q = [ <. J , 1o >. ] ~Q )
41	40	fveq2d 5377	. . . . . . . . 9 |- ( K = J -> ( *Q ` [ <. K , 1o >. ] ~Q ) = ( *Q ` [ <. J , 1o >. ] ~Q ) )
42	41	oveq2d 5742	. . . . . . . 8 |- ( K = J -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) = ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
43	42	breq2d 3905	. . . . . . 7 |- ( K = J -> ( p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <-> p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
44	43	abbidv 2230	. . . . . 6 |- ( K = J -> { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } = { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } )
45	42	breq1d 3903	. . . . . . 7 |- ( K = J -> ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q <-> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q ) )
46	45	abbidv 2230	. . . . . 6 |- ( K = J -> { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } = { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } )
47	44, 46	opeq12d 3677	. . . . 5 |- ( K = J -> <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. = <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. )
48		fveq2 5373	. . . . 5 |- ( K = J -> ( F ` K ) = ( F ` J ) )
49	47, 48	breq12d 3906	. . . 4 |- ( K = J -> ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) <-> <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) ) )
50	49	anbi1d 458	. . 3 |- ( K = J -> ( ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) <-> ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) )
51	50	adantl 273	. 2 |- ( ( ph /\ K = J ) -> ( ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) <-> ( <. { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` J ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) )
52	38, 51	mtbird 645	1 |- ( ( ph /\ K = J ) -> -. ( <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` K ) /\ ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q S } , { q | S <Q q } >. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1312   e. wcel 1461   {cab 2099   A.wral 2388   <.cop 3494   class class class wbr 3893   -->wf 5075   ` cfv 5079  (class class class)co 5726   1oc1o 6258   [cec 6379   N.cnpi 7022   <N clti 7025   ~Q ceq 7029   Q.cnq 7030   +Q cplq 7032   *Qcrq 7034   <Q cltq 7035   P.cnp 7041   +P. cpp 7043   <P cltp 7045

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-1o 6265  df-2o 6266  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-pli 7055  df-mi 7056  df-lti 7057  df-plpq 7094  df-mpq 7095  df-enq 7097  df-nqqs 7098  df-plqqs 7099  df-mqqs 7100  df-1nqqs 7101  df-rq 7102  df-ltnqqs 7103  df-enq0 7174  df-nq0 7175  df-0nq0 7176  df-plq0 7177  df-mq0 7178  df-inp 7216  df-iplp 7218  df-iltp 7220

This theorem is referenced by:  caucvgprprlemnkj  7442