Theorem caucvgsrlemcau 8108

Description: Lemma for caucvgsr 8117. 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 8107	. . . . . . . . . 10 |- ( ph -> G : N. --> P. )
6	5	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> G : N. --> P. )
7		simplr 529	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> n e. N. )
8	6, 7	ffvelcdmd 5813	. . . . . . . 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 5812	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( G ` k ) e. P. )
11		recnnpr 7863	. . . . . . . . . 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 7852	. . . . . . . . 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 8102	. . . . . . . 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 8106	. . . . . . . . 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 8101	. . . . . . . . . 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 8106	. . . . . . . . . . 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 6065	. . . . . . . . 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 2265	. . . . . . . 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 4122	. . . . . . 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 7852	. . . . . . . . 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 8102	. . . . . . . 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 8101	. . . . . . . . . 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 6065	. . . . . . . . 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 2265	. . . . . . . 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 4122	. . . . . . 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 2538	. . 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 2538	. 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 1398   e. wcel 2203   {cab 2218   A.wral 2520   <.cop 3692   class class class wbr 4109   |-> cmpt 4171   -->wf 5348   ` cfv 5352   iota_crio 6002  (class class class)co 6050   1oc1o 6640   [cec 6765   N.cnpi 7587   <N clti 7590   ~Q ceq 7594   *Qcrq 7599   <Q cltq 7600   P.cnp 7606   1Pc1p 7607   +P. cpp 7608   <P cltp 7610   ~R cer 7611   R.cnr 7612   1Rc1r 7614   +R cplr 7616   <R cltr 7618

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-i1p 7782  df-iplp 7783  df-iltp 7785  df-enr 8041  df-nr 8042  df-plr 8043  df-ltr 8045  df-0r 8046  df-1r 8047

This theorem is referenced by:  caucvgsrlemgt1  8110