Theorem caucvgprprlemnkltj 7749

Description: Lemma for caucvgprpr 7772. 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 7656	. . . 4 |- <P Or P.
2		ltrelpr 7565	. . . 4 |- <P C_ ( P. X. P. )
3	1, 2	son2lpi 5062	. . 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 529	. . . . . . 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 7748	. . . . . . . . 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 112	. . . . . . . 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 276	. . . . . . 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 5061	. . . . . . 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 411	. . . . . 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 7679	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
13	12	adantl 277	. . . . . . 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 488	. . . . . . . 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 7607	. . . . . . . 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	ffvelcdmd 5694	. . . . . . . 8 |- ( ph -> ( F ` J ) e. P. )
20	19	ad2antrr 488	. . . . . . 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 7608	. . . . . . . . 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 488	. . . . . . 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 7638	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
26	25	adantl 277	. . . . . . 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 6088	. . . . . 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 167	. . . . 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 7608	. . . . . . . . 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 488	. . . . . . 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 7657	. . . . . . 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 411	. . . . . 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 531	. . . . . 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 5061	. . . . . 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 411	. . . . 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 306	. . . 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 115	. . 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 665	. 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 276	. . . . 5 |- ( ( ph /\ K <N J ) -> S e. Q. )
41		nnnq 7482	. . . . . . 7 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
42		recclnq 7452	. . . . . . 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 276	. . . . 5 |- ( ( ph /\ K <N J ) -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
45		addnqpr 7621	. . . . 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 411	. . . 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 4039	. . 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 465	. 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 674	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   <.cop 3621   class class class wbr 4029   -->wf 5250   ` cfv 5254  (class class class)co 5918   1oc1o 6462   [cec 6585   N.cnpi 7332   <N clti 7335   ~Q ceq 7339   Q.cnq 7340   +Q cplq 7342   *Qcrq 7344   <Q cltq 7345   P.cnp 7351   +P. cpp 7353   <P cltp 7355

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530

This theorem is referenced by:  caucvgprprlemnkj  7752