Theorem caucvgprprlemnkeqj 7604

Description: Lemma for caucvgprpr 7626. 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 7510	. . . 4 |- <P Or P.
2		ltrelpr 7419	. . . 4 |- <P C_ ( P. X. P. )
3	1, 2	son2lpi 4981	. . 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 5602	. . . . . . . 8 |- ( ph -> ( F ` J ) e. P. )
7	6	ad2antrr 480	. . . . . . 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 274	. . . . . . . . . . 11 |- ( ( ph /\ K = J ) -> J e. N. )
9		nnnq 7336	. . . . . . . . . . 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 7306	. . . . . . . . . 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 7461	. . . . . . . . 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 274	. . . . . . 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 7511	. . . . . . 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 409	. . . . . 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 522	. . . . . 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 4980	. . . . . 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 409	. . . . 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 274	. . . . . . . . 9 |- ( ( ph /\ K = J ) -> S e. Q. )
23		nqprlu 7461	. . . . . . . . 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 7511	. . . . . . . 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 409	. . . . . . 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 274	. . . . . 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 521	. . . . . . 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 7475	. . . . . . . . . 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 409	. . . . . . . . 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 3975	. . . . . . . 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 274	. . . . . . 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 4980	. . . . . 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 409	. . . . 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 304	. . . 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 654	. 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 3741	. . . . . . . . . . 11 |- ( K = J -> <. K , 1o >. = <. J , 1o >. )
40	39	eceq1d 6513	. . . . . . . . . 10 |- ( K = J -> [ <. K , 1o >. ] ~Q = [ <. J , 1o >. ] ~Q )
41	40	fveq2d 5471	. . . . . . . . 9 |- ( K = J -> ( *Q ` [ <. K , 1o >. ] ~Q ) = ( *Q ` [ <. J , 1o >. ] ~Q ) )
42	41	oveq2d 5837	. . . . . . . 8 |- ( K = J -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) = ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
43	42	breq2d 3977	. . . . . . 7 |- ( K = J -> ( p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <-> p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
44	43	abbidv 2275	. . . . . 6 |- ( K = J -> { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } = { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } )
45	42	breq1d 3975	. . . . . . 7 |- ( K = J -> ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q <-> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q ) )
46	45	abbidv 2275	. . . . . 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 3749	. . . . 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 5467	. . . . 5 |- ( K = J -> ( F ` K ) = ( F ` J ) )
49	47, 48	breq12d 3978	. . . 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 461	. . 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 275	. 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 663	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 1335   e. wcel 2128   {cab 2143   A.wral 2435   <.cop 3563   class class class wbr 3965   -->wf 5165   ` cfv 5169  (class class class)co 5821   1oc1o 6353   [cec 6475   N.cnpi 7186   <N clti 7189   ~Q ceq 7193   Q.cnq 7194   +Q cplq 7196   *Qcrq 7198   <Q cltq 7199   P.cnp 7205   +P. cpp 7207   <P cltp 7209

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-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-1o 6360  df-2o 6361  df-oadd 6364  df-omul 6365  df-er 6477  df-ec 6479  df-qs 6483  df-ni 7218  df-pli 7219  df-mi 7220  df-lti 7221  df-plpq 7258  df-mpq 7259  df-enq 7261  df-nqqs 7262  df-plqqs 7263  df-mqqs 7264  df-1nqqs 7265  df-rq 7266  df-ltnqqs 7267  df-enq0 7338  df-nq0 7339  df-0nq0 7340  df-plq0 7341  df-mq0 7342  df-inp 7380  df-iplp 7382  df-iltp 7384

This theorem is referenced by:  caucvgprprlemnkj  7606