Theorem caucvgprprlemnbj 8050

Description: Lemma for caucvgprpr 8069. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-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 } >. ) ) ) )
caucvgprprlemnbj.b	|- ( ph -> B e. N. )
caucvgprprlemnbj.j	|- ( ph -> J e. N. )

Assertion

Ref	Expression
caucvgprprlemnbj	|- ( ph -> -. ( ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. ) <P ( F ` J ) )

Distinct variable groups:   B, k, l, n   u, B, k, n   k, F, n   k, J, l, n   u, J

Allowed substitution hints:   ph( u, k, n, l)   F( u, l)

Proof of Theorem caucvgprprlemnbj

Dummy variables p q x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprpr.f	. . . . . . 7 |- ( ph -> F : N. --> P. )
2		caucvgprpr.cau	. . . . . . 7 |- ( 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 } >. ) ) ) )
3	1, 2	caucvgprprlemval 8045	. . . . . 6 |- ( ( ph /\ B <N J ) -> ( ( F ` B ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) /\ ( F ` J ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) ) )
4	3	simprd 114	. . . . 5 |- ( ( ph /\ B <N J ) -> ( F ` J ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) )
5		caucvgprprlemnbj.b	. . . . . . . . 9 |- ( ph -> B e. N. )
6	1, 5	ffvelcdmd 5835	. . . . . . . 8 |- ( ph -> ( F ` B ) e. P. )
7		recnnpr 7905	. . . . . . . . 9 |- ( B e. N. -> <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. e. P. )
8	5, 7	syl 14	. . . . . . . 8 |- ( ph -> <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. e. P. )
9		addclpr 7894	. . . . . . . 8 |- ( ( ( F ` B ) e. P. /\ <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) e. P. )
10	6, 8, 9	syl2anc 415	. . . . . . 7 |- ( ph -> ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) e. P. )
11		caucvgprprlemnbj.j	. . . . . . . 8 |- ( ph -> J e. N. )
12		recnnpr 7905	. . . . . . . 8 |- ( J e. N. -> <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. e. P. )
13	11, 12	syl 14	. . . . . . 7 |- ( ph -> <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. e. P. )
14		ltaddpr 7954	. . . . . . 7 |- ( ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
15	10, 13, 14	syl2anc 415	. . . . . 6 |- ( ph -> ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
16	15	adantr 276	. . . . 5 |- ( ( ph /\ B <N J ) -> ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
17		ltsopr 7953	. . . . . 6 |- <P Or P.
18		ltrelpr 7862	. . . . . 6 |- <P C_ ( P. X. P. )
19	17, 18	sotri 5178	. . . . 5 |- ( ( ( F ` J ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) /\ ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) ) -> ( F ` J ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
20	4, 16, 19	syl2anc 415	. . . 4 |- ( ( ph /\ B <N J ) -> ( F ` J ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
21		ltaddpr 7954	. . . . . . . 8 |- ( ( ( F ` B ) e. P. /\ <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( F ` B ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) )
22	6, 8, 21	syl2anc 415	. . . . . . 7 |- ( ph -> ( F ` B ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) )
23	22	adantr 276	. . . . . 6 |- ( ( ph /\ B = J ) -> ( F ` B ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) )
24		fveq2 5690	. . . . . . . 8 |- ( B = J -> ( F ` B ) = ( F ` J ) )
25	24	breq1d 4135	. . . . . . 7 |- ( B = J -> ( ( F ` B ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) <-> ( F ` J ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) ) )
26	25	adantl 277	. . . . . 6 |- ( ( ph /\ B = J ) -> ( ( F ` B ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) <-> ( F ` J ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) ) )
27	23, 26	mpbid 147	. . . . 5 |- ( ( ph /\ B = J ) -> ( F ` J ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) )
28	15	adantr 276	. . . . 5 |- ( ( ph /\ B = J ) -> ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
29	27, 28, 19	syl2anc 415	. . . 4 |- ( ( ph /\ B = J ) -> ( F ` J ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
30	1, 2	caucvgprprlemval 8045	. . . . . 6 |- ( ( ph /\ J <N B ) -> ( ( F ` J ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) /\ ( F ` B ) <P ( ( F ` J ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) ) )
31	30	simpld 112	. . . . 5 |- ( ( ph /\ J <N B ) -> ( F ` J ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
32		ltaprg 7976	. . . . . . . . 9 |- ( ( x e. P. /\ y e. P. /\ z e. P. ) -> ( x <P y <-> ( z +P. x ) <P ( z +P. y ) ) )
33	32	adantl 277	. . . . . . . 8 |- ( ( ph /\ ( x e. P. /\ y e. P. /\ z e. P. ) ) -> ( x <P y <-> ( z +P. x ) <P ( z +P. y ) ) )
34		addcomprg 7935	. . . . . . . . 9 |- ( ( x e. P. /\ y e. P. ) -> ( x +P. y ) = ( y +P. x ) )
35	34	adantl 277	. . . . . . . 8 |- ( ( ph /\ ( x e. P. /\ y e. P. ) ) -> ( x +P. y ) = ( y +P. x ) )
36	33, 6, 10, 13, 35	caovord2d 6249	. . . . . . 7 |- ( ph -> ( ( F ` B ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) <-> ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) ) )
37	22, 36	mpbid 147	. . . . . 6 |- ( ph -> ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
38	37	adantr 276	. . . . 5 |- ( ( ph /\ J <N B ) -> ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
39	17, 18	sotri 5178	. . . . 5 |- ( ( ( F ` J ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) /\ ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) ) -> ( F ` J ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
40	31, 38, 39	syl2anc 415	. . . 4 |- ( ( ph /\ J <N B ) -> ( F ` J ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
41		pitri3or 7679	. . . . 5 |- ( ( B e. N. /\ J e. N. ) -> ( B <N J \/ B = J \/ J <N B ) )
42	5, 11, 41	syl2anc 415	. . . 4 |- ( ph -> ( B <N J \/ B = J \/ J <N B ) )
43	20, 29, 40, 42	mpjao3dan 1348	. . 3 |- ( ph -> ( F ` J ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) )
44	1, 11	ffvelcdmd 5835	. . . . 5 |- ( ph -> ( F ` J ) e. P. )
45		addclpr 7894	. . . . . 6 |- ( ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) e. P. )
46	10, 13, 45	syl2anc 415	. . . . 5 |- ( ph -> ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) e. P. )
47		so2nr 4461	. . . . . 6 |- ( ( <P Or P. /\ ( ( F ` J ) e. P. /\ ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) e. P. ) ) -> -. ( ( F ` J ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) /\ ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` J ) ) )
48	17, 47	mpan 428	. . . . 5 |- ( ( ( F ` J ) e. P. /\ ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) e. P. ) -> -. ( ( F ` J ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) /\ ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` J ) ) )
49	44, 46, 48	syl2anc 415	. . . 4 |- ( ph -> -. ( ( F ` J ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) /\ ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` J ) ) )
50		imnan 701	. . . 4 |- ( ( ( F ` J ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) -> -. ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` J ) ) <-> -. ( ( F ` J ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) /\ ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` J ) ) )
51	49, 50	sylibr 134	. . 3 |- ( ph -> ( ( F ` J ) <P ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) -> -. ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` J ) ) )
52	43, 51	mpd 13	. 2 |- ( ph -> -. ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` J ) )
53		breq1 4128	. . . . . . 7 |- ( p = l -> ( p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) )
54	53	cbvabv 2365	. . . . . 6 |- { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) }
55		breq2 4129	. . . . . . 7 |- ( q = u -> ( ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u ) )
56	55	cbvabv 2365	. . . . . 6 |- { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } = { u | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u }
57	54, 56	opeq12i 3904	. . . . 5 |- <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. = <. { l | l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u } >.
58	57	oveq2i 6086	. . . 4 |- ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u } >. )
59		breq1 4128	. . . . . 6 |- ( p = l -> ( p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
60	59	cbvabv 2365	. . . . 5 |- { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) }
61		breq2 4129	. . . . . 6 |- ( q = u -> ( ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u ) )
62	61	cbvabv 2365	. . . . 5 |- { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } = { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u }
63	60, 62	opeq12i 3904	. . . 4 |- <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. = <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >.
64	58, 63	oveq12i 6087	. . 3 |- ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) = ( ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. )
65	64	breq1i 4132	. 2 |- ( ( ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` J ) <-> ( ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. ) <P ( F ` J ) )
66	52, 65	sylnib 687	1 |- ( ph -> -. ( ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u } >. ) +P. <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. ) <P ( F ` J ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1008   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   <.cop 3708   class class class wbr 4125   Or wor 4435   -->wf 5368   ` cfv 5372  (class class class)co 6075   1oc1o 6670   [cec 6795   N.cnpi 7629   <N clti 7632   ~Q ceq 7636   *Qcrq 7641   <Q cltq 7642   P.cnp 7648   +P. cpp 7650   <P cltp 7652

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iplp 7825  df-iltp 7827

This theorem is referenced by:  caucvgprprlemaddq  8065