Theorem caucvgprprlemnbj 7667

Description: Lemma for caucvgprpr 7686. 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 7662	. . . . . 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 5644	. . . . . . . 8 |- ( ph -> ( F ` B ) e. P. )
7		recnnpr 7522	. . . . . . . . 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 7511	. . . . . . . 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 411	. . . . . . 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 7522	. . . . . . . 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 7571	. . . . . . 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 411	. . . . . 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 7570	. . . . . 6 |- <P Or P.
18		ltrelpr 7479	. . . . . 6 |- <P C_ ( P. X. P. )
19	17, 18	sotri 5016	. . . . 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 411	. . . 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 7571	. . . . . . . 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 411	. . . . . . 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 5507	. . . . . . . 8 |- ( B = J -> ( F ` B ) = ( F ` J ) )
25	24	breq1d 4008	. . . . . . 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 411	. . . 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 7662	. . . . . 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 7593	. . . . . . . . 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 7552	. . . . . . . . 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 6034	. . . . . . 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 5016	. . . . 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 411	. . . 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 7296	. . . . 5 |- ( ( B e. N. /\ J e. N. ) -> ( B <N J \/ B = J \/ J <N B ) )
42	5, 11, 41	syl2anc 411	. . . 4 |- ( ph -> ( B <N J \/ B = J \/ J <N B ) )
43	20, 29, 40, 42	mpjao3dan 1307	. . 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 5644	. . . . 5 |- ( ph -> ( F ` J ) e. P. )
45		addclpr 7511	. . . . . 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 411	. . . . 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 4315	. . . . . 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 424	. . . . 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 411	. . . 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 690	. . . 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 4001	. . . . . . 7 |- ( p = l -> ( p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) )
54	53	cbvabv 2300	. . . . . 6 |- { p | p <Q ( *Q ` [ <. B , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. B , 1o >. ] ~Q ) }
55		breq2 4002	. . . . . . 7 |- ( q = u -> ( ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u ) )
56	55	cbvabv 2300	. . . . . 6 |- { q | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q q } = { u | ( *Q ` [ <. B , 1o >. ] ~Q ) <Q u }
57	54, 56	opeq12i 3779	. . . . 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 5876	. . . 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 4001	. . . . . 6 |- ( p = l -> ( p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
60	59	cbvabv 2300	. . . . 5 |- { p | p <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) }
61		breq2 4002	. . . . . 6 |- ( q = u -> ( ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u ) )
62	61	cbvabv 2300	. . . . 5 |- { q | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q q } = { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u }
63	60, 62	opeq12i 3779	. . . 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 5877	. . 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 4005	. 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 676	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 977   /\ w3a 978   = wceq 1353   e. wcel 2146   {cab 2161   A.wral 2453   <.cop 3592   class class class wbr 3998   Or wor 4289   -->wf 5204   ` cfv 5208  (class class class)co 5865   1oc1o 6400   [cec 6523   N.cnpi 7246   <N clti 7249   ~Q ceq 7253   *Qcrq 7258   <Q cltq 7259   P.cnp 7265   +P. cpp 7267   <P cltp 7269

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-2o 6408  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-enq0 7398  df-nq0 7399  df-0nq0 7400  df-plq0 7401  df-mq0 7402  df-inp 7440  df-iplp 7442  df-iltp 7444

This theorem is referenced by:  caucvgprprlemaddq  7682