Theorem caucvgprprlemnkltj 6879

Description: Lemma for caucvgprpr 6902. 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
caucvgprprlemnkltj	|- ( ( ph /\ K <N 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   K, l, p   u, K, q   S, p, q   k, l, n   u, k, n

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( k, n)

Proof of Theorem caucvgprprlemnkltj

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltsopr 6786	. . . 4 |- <P Or P.
2		ltrelpr 6695	. . . 4 |- <P C_ ( P. X. P. )
3	1, 2	son2lpi 4741	. . 3 |- -. ( <. { p | p <Q S } , { q | S <Q q } >. <P ( F ` J ) /\ ( F ` J ) <P <. { p | p <Q S } , { q | S <Q q } >. )
4		simprl 497	. . . . . . 7 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` K ) )
5		caucvgprpr.f	. . . . . . . . . 10 |- ( ph -> F : N. --> P. )
6		caucvgprpr.cau	. . . . . . . . . 10 |- ( 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 } >. ) ) ) )
7	5, 6	caucvgprprlemval 6878	. . . . . . . . 9 |- ( ( ph /\ K <N J ) -> ( ( F ` K ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) /\ ( F ` J ) <P ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) ) )
8	7	simpld 110	. . . . . . . 8 |- ( ( ph /\ K <N J ) -> ( F ` K ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) )
9	8	adantr 270	. . . . . . 7 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 } >. ) ) -> ( F ` K ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) )
10	1, 2	sotri 4740	. . . . . . 7 |- ( ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` K ) /\ ( F ` K ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) ) -> ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) )
11	4, 9, 10	syl2anc 403	. . . . . 6 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) )
12		ltaprg 6809	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
13	12	adantl 271	. . . . . . 7 |- ( ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 } >. ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
14		caucvgprprlemnkj.s	. . . . . . . . 9 |- ( ph -> S e. Q. )
15	14	ad2antrr 471	. . . . . . . 8 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 } >. ) ) -> S e. Q. )
16		nqprlu 6737	. . . . . . . 8 |- ( S e. Q. -> <. { p | p <Q S } , { q | S <Q q } >. e. P. )
17	15, 16	syl 14	. . . . . . 7 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 | S <Q q } >. e. P. )
18		caucvgprprlemnkj.j	. . . . . . . . 9 |- ( ph -> J e. N. )
19	5, 18	ffvelrnd 5324	. . . . . . . 8 |- ( ph -> ( F ` J ) e. P. )
20	19	ad2antrr 471	. . . . . . 7 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 } >. ) ) -> ( F ` J ) e. P. )
21		caucvgprprlemnkj.k	. . . . . . . . 9 |- ( ph -> K e. N. )
22		recnnpr 6738	. . . . . . . . 9 |- ( K e. N. -> <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. e. P. )
23	21, 22	syl 14	. . . . . . . 8 |- ( ph -> <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. e. P. )
24	23	ad2antrr 471	. . . . . . 7 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. e. P. )
25		addcomprg 6768	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
26	25	adantl 271	. . . . . . 7 |- ( ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 } >. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
27	13, 17, 20, 24, 26	caovord2d 5690	. . . . . 6 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 | S <Q q } >. <P ( F ` J ) <-> ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) ) )
28	11, 27	mpbird 165	. . . . 5 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 | S <Q q } >. <P ( F ` J ) )
29		recnnpr 6738	. . . . . . . . 9 |- ( J e. N. -> <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. e. P. )
30	18, 29	syl 14	. . . . . . . 8 |- ( ph -> <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. e. P. )
31	30	ad2antrr 471	. . . . . . 7 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. e. P. )
32		ltaddpr 6787	. . . . . . 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 } >. ) )
33	20, 31, 32	syl2anc 403	. . . . . 6 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 } >. ) ) -> ( F ` J ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
34		simprr 498	. . . . . 6 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 } >. ) ) -> ( ( 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 } >. )
35	1, 2	sotri 4740	. . . . . 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 } >. )
36	33, 34, 35	syl2anc 403	. . . . 5 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 } >. ) ) -> ( F ` J ) <P <. { p | p <Q S } , { q | S <Q q } >. )
37	28, 36	jca 300	. . . 4 |- ( ( ( ph /\ K <N J ) /\ ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 | S <Q q } >. <P ( F ` J ) /\ ( F ` J ) <P <. { p | p <Q S } , { q | S <Q q } >. ) )
38	37	ex 113	. . 3 |- ( ( ph /\ K <N J ) -> ( ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 | S <Q q } >. <P ( F ` J ) /\ ( F ` J ) <P <. { p | p <Q S } , { q | S <Q q } >. ) ) )
39	3, 38	mtoi 622	. 2 |- ( ( ph /\ K <N J ) -> -. ( ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 } >. ) )
40	14	adantr 270	. . . . 5 |- ( ( ph /\ K <N J ) -> S e. Q. )
41		nnnq 6612	. . . . . . 7 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
42		recclnq 6582	. . . . . . 7 |- ( [ <. K , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
43	21, 41, 42	3syl 17	. . . . . 6 |- ( ph -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
44	43	adantr 270	. . . . 5 |- ( ( ph /\ K <N J ) -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
45		addnqpr 6751	. . . . 5 |- ( ( S e. Q. /\ ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. ) -> <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. = ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) )
46	40, 44, 45	syl2anc 403	. . . 4 |- ( ( ph /\ K <N J ) -> <. { p | p <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) } , { q | ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q q } >. = ( <. { p | p <Q S } , { q | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) )
47	46	breq1d 3795	. . 3 |- ( ( ph /\ K <N 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 | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` K ) ) )
48	47	anbi1d 452	. 2 |- ( ( ph /\ K <N 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 | S <Q q } >. +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { 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 } >. ) ) )
49	39, 48	mtbird 630	1 |- ( ( ph /\ K <N 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 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   <.cop 3401   class class class wbr 3785   -->wf 4918   ` cfv 4922  (class class class)co 5532   1oc1o 6017   [cec 6127   N.cnpi 6462   <N clti 6465   ~Q ceq 6469   Q.cnq 6470   +Q cplq 6472   *Qcrq 6474   <Q cltq 6475   P.cnp 6481   +P. cpp 6483   <P cltp 6485

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-iplp 6658  df-iltp 6660

This theorem is referenced by:  caucvgprprlemnkj  6882