Theorem caucvgprprlemnkeqj 7691

Description: Lemma for caucvgprpr 7713. 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 7597	. . . 4 |- <P Or P.
2		ltrelpr 7506	. . . 4 |- <P C_ ( P. X. P. )
3	1, 2	son2lpi 5027	. . 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	ffvelcdmd 5654	. . . . . . . 8 |- ( ph -> ( F ` J ) e. P. )
7	6	ad2antrr 488	. . . . . . 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 276	. . . . . . . . . . 11 |- ( ( ph /\ K = J ) -> J e. N. )
9		nnnq 7423	. . . . . . . . . . 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 7393	. . . . . . . . . 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 7548	. . . . . . . . 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 276	. . . . . . 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 7598	. . . . . . 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 411	. . . . . 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 531	. . . . . 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 5026	. . . . . 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 411	. . . . 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 276	. . . . . . . . 9 |- ( ( ph /\ K = J ) -> S e. Q. )
23		nqprlu 7548	. . . . . . . . 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 7598	. . . . . . . 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 411	. . . . . . 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 276	. . . . . 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 529	. . . . . . 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 7562	. . . . . . . . . 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 411	. . . . . . . . 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 4015	. . . . . . . 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 276	. . . . . . 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 147	. . . . . 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 5026	. . . . . 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 411	. . . . 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 306	. . . 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 115	. . 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 664	. 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 3780	. . . . . . . . . . 11 |- ( K = J -> <. K , 1o >. = <. J , 1o >. )
40	39	eceq1d 6573	. . . . . . . . . 10 |- ( K = J -> [ <. K , 1o >. ] ~Q = [ <. J , 1o >. ] ~Q )
41	40	fveq2d 5521	. . . . . . . . 9 |- ( K = J -> ( *Q ` [ <. K , 1o >. ] ~Q ) = ( *Q ` [ <. J , 1o >. ] ~Q ) )
42	41	oveq2d 5893	. . . . . . . 8 |- ( K = J -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) = ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
43	42	breq2d 4017	. . . . . . 7 |- ( K = J -> ( p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <-> p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
44	43	abbidv 2295	. . . . . 6 |- ( K = J -> { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } = { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } )
45	42	breq1d 4015	. . . . . . 7 |- ( K = J -> ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q <-> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q ) )
46	45	abbidv 2295	. . . . . 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 3788	. . . . 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 5517	. . . . 5 |- ( K = J -> ( F ` K ) = ( F ` J ) )
49	47, 48	breq12d 4018	. . . 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 465	. . 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 277	. 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 673	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 104   <-> wb 105   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   <.cop 3597   class class class wbr 4005   -->wf 5214   ` cfv 5218  (class class class)co 5877   1oc1o 6412   [cec 6535   N.cnpi 7273   <N clti 7276   ~Q ceq 7280   Q.cnq 7281   +Q cplq 7283   *Qcrq 7285   <Q cltq 7286   P.cnp 7292   +P. cpp 7294   <P cltp 7296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-iplp 7469  df-iltp 7471

This theorem is referenced by:  caucvgprprlemnkj  7693