Theorem caucvgsrlemcau 7067

Description: Lemma for caucvgsr 7076. 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 7066	. . . . . . . . . 10 |- ( ph -> G : N. --> P. )
6	5	ad2antrr 472	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> G : N. --> P. )
7		simplr 497	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> n e. N. )
8	6, 7	ffvelrnd 5356	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( G ` n ) e. P. )
9	5	adantr 270	. . . . . . . . . 10 |- ( ( ph /\ n e. N. ) -> G : N. --> P. )
10	9	ffvelrnda 5355	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( G ` k ) e. P. )
11		recnnpr 6836	. . . . . . . . . 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 6825	. . . . . . . . 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 403	. . . . . . . 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 7061	. . . . . . . 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 403	. . . . . . 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 7065	. . . . . . . . 9 |- ( ( ph /\ n e. N. ) -> [ <. ( ( G ` n ) +P. 1P ) , 1P >. ] ~R = ( F ` n ) )
18	17	adantr 270	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> [ <. ( ( G ` n ) +P. 1P ) , 1P >. ] ~R = ( F ` n ) )
19		prsradd 7060	. . . . . . . . . 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 403	. . . . . . . . 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 7065	. . . . . . . . . . 11 |- ( ( ph /\ k e. N. ) -> [ <. ( ( G ` k ) +P. 1P ) , 1P >. ] ~R = ( F ` k ) )
22	21	adantlr 461	. . . . . . . . . 10 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> [ <. ( ( G ` k ) +P. 1P ) , 1P >. ] ~R = ( F ` k ) )
23	22	oveq1d 5579	. . . . . . . . 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 2115	. . . . . . . 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 3819	. . . . . . 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 186	. . . . . 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 6825	. . . . . . . . 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 403	. . . . . . . 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 7061	. . . . . . . 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 403	. . . . . . 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 7060	. . . . . . . . . 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 403	. . . . . . . . 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 5579	. . . . . . . . 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 2115	. . . . . . . 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 3819	. . . . . . 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 186	. . . . . 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 457	. . . . 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 228	. . . 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 2369	. . 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 2369	. 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 165	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 102   <-> wb 103   = wceq 1285   e. wcel 1434   {cab 2069   A.wral 2353   <.cop 3420   class class class wbr 3806   |-> cmpt 3860   -->wf 4949   ` cfv 4953   iota_crio 5519  (class class class)co 5564   1oc1o 6079   [cec 6192   N.cnpi 6560   <N clti 6563   ~Q ceq 6567   *Qcrq 6572   <Q cltq 6573   P.cnp 6579   1Pc1p 6580   +P. cpp 6581   <P cltp 6583   ~R cer 6584   R.cnr 6585   1Rc1r 6587   +R cplr 6589   <R cltr 6591

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-eprel 4073  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-2o 6087  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6592  df-pli 6593  df-mi 6594  df-lti 6595  df-plpq 6632  df-mpq 6633  df-enq 6635  df-nqqs 6636  df-plqqs 6637  df-mqqs 6638  df-1nqqs 6639  df-rq 6640  df-ltnqqs 6641  df-enq0 6712  df-nq0 6713  df-0nq0 6714  df-plq0 6715  df-mq0 6716  df-inp 6754  df-i1p 6755  df-iplp 6756  df-iltp 6758  df-enr 7001  df-nr 7002  df-plr 7003  df-ltr 7005  df-0r 7006  df-1r 7007

This theorem is referenced by:  caucvgsrlemgt1  7069