Theorem caucvgprprlemnkeqj 7802

Description: Lemma for caucvgprpr 7824. 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 7708	. . . 4 |- <P Or P.
2		ltrelpr 7617	. . . 4 |- <P C_ ( P. X. P. )
3	1, 2	son2lpi 5078	. . 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 5715	. . . . . . . 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 7534	. . . . . . . . . . 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 7504	. . . . . . . . . 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 7659	. . . . . . . . 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 7709	. . . . . . 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 5077	. . . . . 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 7659	. . . . . . . . 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 7709	. . . . . . . 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 7673	. . . . . . . . . 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 4053	. . . . . . . 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 5077	. . . . . 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 665	. 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 3818	. . . . . . . . . . 11 |- ( K = J -> <. K , 1o >. = <. J , 1o >. )
40	39	eceq1d 6655	. . . . . . . . . 10 |- ( K = J -> [ <. K , 1o >. ] ~Q = [ <. J , 1o >. ] ~Q )
41	40	fveq2d 5579	. . . . . . . . 9 |- ( K = J -> ( *Q ` [ <. K , 1o >. ] ~Q ) = ( *Q ` [ <. J , 1o >. ] ~Q ) )
42	41	oveq2d 5959	. . . . . . . 8 |- ( K = J -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) = ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
43	42	breq2d 4055	. . . . . . 7 |- ( K = J -> ( p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <-> p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
44	43	abbidv 2322	. . . . . 6 |- ( K = J -> { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } = { p | p <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) } )
45	42	breq1d 4053	. . . . . . 7 |- ( K = J -> ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q <-> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q q ) )
46	45	abbidv 2322	. . . . . 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 3826	. . . . 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 5575	. . . . 5 |- ( K = J -> ( F ` K ) = ( F ` J ) )
49	47, 48	breq12d 4056	. . . 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 674	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 1372   e. wcel 2175   {cab 2190   A.wral 2483   <.cop 3635   class class class wbr 4043   -->wf 5266   ` cfv 5270  (class class class)co 5943   1oc1o 6494   [cec 6617   N.cnpi 7384   <N clti 7387   ~Q ceq 7391   Q.cnq 7392   +Q cplq 7394   *Qcrq 7396   <Q cltq 7397   P.cnp 7403   +P. cpp 7405   <P cltp 7407

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-2o 6502  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-pli 7417  df-mi 7418  df-lti 7419  df-plpq 7456  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-plqqs 7461  df-mqqs 7462  df-1nqqs 7463  df-rq 7464  df-ltnqqs 7465  df-enq0 7536  df-nq0 7537  df-0nq0 7538  df-plq0 7539  df-mq0 7540  df-inp 7578  df-iplp 7580  df-iltp 7582

This theorem is referenced by:  caucvgprprlemnkj  7804