Theorem caucvgprprlemnkeqj 7873

Description: Lemma for caucvgprpr 7895. 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 7779	. . . 4 |- <P Or P.
2		ltrelpr 7688	. . . 4 |- <P C_ ( P. X. P. )
3	1, 2	son2lpi 5124	. . 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 5770	. . . . . . . 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 7605	. . . . . . . . . . 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 7575	. . . . . . . . . 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 7730	. . . . . . . . 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 7780	. . . . . . 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 5123	. . . . . 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 7730	. . . . . . . . 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 7780	. . . . . . . 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 7744	. . . . . . . . . 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 4092	. . . . . . . 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 5123	. . . . . 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 668	. 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 3856	. . . . . . . . . . 11 |- ( K = J -> <. K , 1o >. = <. J , 1o >. )
40	39	eceq1d 6714	. . . . . . . . . 10 |- ( K = J -> [ <. K , 1o >. ] ~Q = [ <. J , 1o >. ] ~Q )
41	40	fveq2d 5630	. . . . . . . . 9 |- ( K = J -> ( *Q ` [ <. K , 1o >. ] ~Q ) = ( *Q ` [ <. J , 1o >. ] ~Q ) )
42	41	oveq2d 6016	. . . . . . . 8 |- ( K = J -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) = ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
43	42	breq2d 4094	. . . . . . 7 |- ( K = J -> ( p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <-> p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
44	43	abbidv 2347	. . . . . 6 |- ( K = J -> { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } = { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } )
45	42	breq1d 4092	. . . . . . 7 |- ( K = J -> ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q <-> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q ) )
46	45	abbidv 2347	. . . . . 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 3864	. . . . 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 5626	. . . . 5 |- ( K = J -> ( F ` K ) = ( F ` J ) )
49	47, 48	breq12d 4095	. . . 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 677	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 1395   e. wcel 2200   {cab 2215   A.wral 2508   <.cop 3669   class class class wbr 4082   -->wf 5313   ` cfv 5317  (class class class)co 6000   1oc1o 6553   [cec 6676   N.cnpi 7455   <N clti 7458   ~Q ceq 7462   Q.cnq 7463   +Q cplq 7465   *Qcrq 7467   <Q cltq 7468   P.cnp 7474   +P. cpp 7476   <P cltp 7478

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-2o 6561  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-enq0 7607  df-nq0 7608  df-0nq0 7609  df-plq0 7610  df-mq0 7611  df-inp 7649  df-iplp 7651  df-iltp 7653

This theorem is referenced by:  caucvgprprlemnkj  7875