Theorem caucvgsrlemcau 7976

Description: Lemma for caucvgsr 7985. 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 7975	. . . . . . . . . 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 5770	. . . . . . . 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 5769	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( G ` k ) e. P. )
11		recnnpr 7731	. . . . . . . . . 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 7720	. . . . . . . . 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 7970	. . . . . . . 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 7974	. . . . . . . . 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 7969	. . . . . . . . . 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 7974	. . . . . . . . . . 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 6015	. . . . . . . . 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 2262	. . . . . . . 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 4095	. . . . . . 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 7720	. . . . . . . . 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 7970	. . . . . . . 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 7969	. . . . . . . . . 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 6015	. . . . . . . . 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 2262	. . . . . . . 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 4095	. . . . . . 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 2526	. . 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 2526	. 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 1395   e. wcel 2200   {cab 2215   A.wral 2508   <.cop 3669   class class class wbr 4082   |-> cmpt 4144   -->wf 5313   ` cfv 5317   iota_crio 5952  (class class class)co 6000   1oc1o 6553   [cec 6676   N.cnpi 7455   <N clti 7458   ~Q ceq 7462   *Qcrq 7467   <Q cltq 7468   P.cnp 7474   1Pc1p 7475   +P. cpp 7476   <P cltp 7478   ~R cer 7479   R.cnr 7480   1Rc1r 7482   +R cplr 7484   <R cltr 7486

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-2o 6561  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-enq0 7607  df-nq0 7608  df-0nq0 7609  df-plq0 7610  df-mq0 7611  df-inp 7649  df-i1p 7650  df-iplp 7651  df-iltp 7653  df-enr 7909  df-nr 7910  df-plr 7911  df-ltr 7913  df-0r 7914  df-1r 7915

This theorem is referenced by:  caucvgsrlemgt1  7978