Theorem caucvgsrlemcau 7905

Description: Lemma for caucvgsr 7914. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)

Hypotheses

Ref	Expression
caucvgsr.f	|- ( ph -> F : N. --> R. )
caucvgsr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
caucvgsrlemgt1.gt1	|- ( ph -> A. m e. N. 1R <R ( F ` m ) )
caucvgsrlemf.xfr	|- G = ( x e. N. |-> ( iota_ y e. P. ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )

Assertion

Ref	Expression
caucvgsrlemcau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <P ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( G ` k ) <P ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )

Distinct variable groups:   m, F, x   y, F, x   k, m, n, x   ph, k, n, x   y, k, n   n, l, u

Allowed substitution hints:   ph( y, u, m, l)   F( u, k, n, l)   G( x, y, u, k, m, n, l)

Proof of Theorem caucvgsrlemcau

Step	Hyp	Ref	Expression
1		caucvgsr.cau	. 2 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
2		caucvgsr.f	. . . . . . . . . . 11 |- ( ph -> F : N. --> R. )
3		caucvgsrlemgt1.gt1	. . . . . . . . . . 11 |- ( ph -> A. m e. N. 1R <R ( F ` m ) )
4		caucvgsrlemf.xfr	. . . . . . . . . . 11 |- G = ( x e. N. |-> ( iota_ y e. P. ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )
5	2, 1, 3, 4	caucvgsrlemf 7904	. . . . . . . . . 10 |- ( ph -> G : N. --> P. )
6	5	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> G : N. --> P. )
7		simplr 528	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> n e. N. )
8	6, 7	ffvelcdmd 5715	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( G ` n ) e. P. )
9	5	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ n e. N. ) -> G : N. --> P. )
10	9	ffvelcdmda 5714	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( G ` k ) e. P. )
11		recnnpr 7660	. . . . . . . . . 10 |- ( n e. N. -> <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. e. P. )
12	7, 11	syl 14	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. e. P. )
13		addclpr 7649	. . . . . . . . 9 |- ( ( ( G ` k ) e. P. /\ <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. e. P. ) -> ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) e. P. )
14	10, 12, 13	syl2anc 411	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) e. P. )
15		prsrlt 7899	. . . . . . . 8 |- ( ( ( G ` n ) e. P. /\ ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) e. P. ) -> ( ( G ` n ) <P ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) <-> [ <. ( ( G ` n ) +P. 1P ) , 1P >. ] ~R <R [ <. ( ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) +P. 1P ) , 1P >. ] ~R ) )
16	8, 14, 15	syl2anc 411	. . . . . . 7 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( G ` n ) <P ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) <-> [ <. ( ( G ` n ) +P. 1P ) , 1P >. ] ~R <R [ <. ( ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) +P. 1P ) , 1P >. ] ~R ) )
17	2, 1, 3, 4	caucvgsrlemfv 7903	. . . . . . . . 9 |- ( ( ph /\ n e. N. ) -> [ <. ( ( G ` n ) +P. 1P ) , 1P >. ] ~R = ( F ` n ) )
18	17	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> [ <. ( ( G ` n ) +P. 1P ) , 1P >. ] ~R = ( F ` n ) )
19		prsradd 7898	. . . . . . . . . 10 |- ( ( ( G ` k ) e. P. /\ <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. e. P. ) -> [ <. ( ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) +P. 1P ) , 1P >. ] ~R = ( [ <. ( ( G ` k ) +P. 1P ) , 1P >. ] ~R +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
20	10, 12, 19	syl2anc 411	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> [ <. ( ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) +P. 1P ) , 1P >. ] ~R = ( [ <. ( ( G ` k ) +P. 1P ) , 1P >. ] ~R +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
21	2, 1, 3, 4	caucvgsrlemfv 7903	. . . . . . . . . . 11 |- ( ( ph /\ k e. N. ) -> [ <. ( ( G ` k ) +P. 1P ) , 1P >. ] ~R = ( F ` k ) )
22	21	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> [ <. ( ( G ` k ) +P. 1P ) , 1P >. ] ~R = ( F ` k ) )
23	22	oveq1d 5958	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( [ <. ( ( G ` k ) +P. 1P ) , 1P >. ] ~R +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) = ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
24	20, 23	eqtrd 2237	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> [ <. ( ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) +P. 1P ) , 1P >. ] ~R = ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
25	18, 24	breq12d 4056	. . . . . . 7 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( [ <. ( ( G ` n ) +P. 1P ) , 1P >. ] ~R <R [ <. ( ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) +P. 1P ) , 1P >. ] ~R <-> ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
26	16, 25	bitrd 188	. . . . . 6 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( G ` n ) <P ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) <-> ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
27		addclpr 7649	. . . . . . . . 9 |- ( ( ( G ` n ) e. P. /\ <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. e. P. ) -> ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) e. P. )
28	8, 12, 27	syl2anc 411	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) e. P. )
29		prsrlt 7899	. . . . . . . 8 |- ( ( ( G ` k ) e. P. /\ ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) e. P. ) -> ( ( G ` k ) <P ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) <-> [ <. ( ( G ` k ) +P. 1P ) , 1P >. ] ~R <R [ <. ( ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) +P. 1P ) , 1P >. ] ~R ) )
30	10, 28, 29	syl2anc 411	. . . . . . 7 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( G ` k ) <P ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) <-> [ <. ( ( G ` k ) +P. 1P ) , 1P >. ] ~R <R [ <. ( ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) +P. 1P ) , 1P >. ] ~R ) )
31		prsradd 7898	. . . . . . . . . 10 |- ( ( ( G ` n ) e. P. /\ <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. e. P. ) -> [ <. ( ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) +P. 1P ) , 1P >. ] ~R = ( [ <. ( ( G ` n ) +P. 1P ) , 1P >. ] ~R +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
32	8, 12, 31	syl2anc 411	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> [ <. ( ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) +P. 1P ) , 1P >. ] ~R = ( [ <. ( ( G ` n ) +P. 1P ) , 1P >. ] ~R +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
33	18	oveq1d 5958	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( [ <. ( ( G ` n ) +P. 1P ) , 1P >. ] ~R +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) = ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
34	32, 33	eqtrd 2237	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> [ <. ( ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) +P. 1P ) , 1P >. ] ~R = ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
35	22, 34	breq12d 4056	. . . . . . 7 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( [ <. ( ( G ` k ) +P. 1P ) , 1P >. ] ~R <R [ <. ( ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) +P. 1P ) , 1P >. ] ~R <-> ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
36	30, 35	bitrd 188	. . . . . 6 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( G ` k ) <P ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) <-> ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
37	26, 36	anbi12d 473	. . . . 5 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( G ` n ) <P ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( G ` k ) <P ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) <-> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
38	37	imbi2d 230	. . . 4 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( n <N k -> ( ( G ` n ) <P ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( G ` k ) <P ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) <-> ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) ) )
39	38	ralbidva 2501	. . 3 |- ( ( ph /\ n e. N. ) -> ( A. k e. N. ( n <N k -> ( ( G ` n ) <P ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( G ` k ) <P ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) <-> A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) ) )
40	39	ralbidva 2501	. 2 |- ( ph -> ( A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <P ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( G ` k ) <P ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) <-> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) ) )
41	1, 40	mpbird 167	1 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <P ( ( G ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( G ` k ) <P ( ( G ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   {cab 2190   A.wral 2483   <.cop 3635   class class class wbr 4043   |-> cmpt 4104   -->wf 5266   ` cfv 5270   iota_crio 5897  (class class class)co 5943   1oc1o 6494   [cec 6617   N.cnpi 7384   <N clti 7387   ~Q ceq 7391   *Qcrq 7396   <Q cltq 7397   P.cnp 7403   1Pc1p 7404   +P. cpp 7405   <P cltp 7407   ~R cer 7408   R.cnr 7409   1Rc1r 7411   +R cplr 7413   <R cltr 7415

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-rmo 2491  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-riota 5898  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-i1p 7579  df-iplp 7580  df-iltp 7582  df-enr 7838  df-nr 7839  df-plr 7840  df-ltr 7842  df-0r 7843  df-1r 7844

This theorem is referenced by:  caucvgsrlemgt1  7907