Theorem caucvgsrlemoffcau 7799

Description: Lemma for caucvgsr 7803. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-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 ) ) ) )
caucvgsrlembnd.bnd	|- ( ph -> A. m e. N. A <R ( F ` m ) )
caucvgsrlembnd.offset	|- G = ( a e. N. |-> ( ( ( F ` a ) +R 1R ) +R ( A .R -1R ) ) )

Assertion

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

Distinct variable groups:   A, a   A, m   F, a   k, a, n, ph   n, l, u

Allowed substitution hints:   ph( u, m, l)   A( u, k, n, l)   F( u, k, m, n, l)   G( u, k, m, n, a, l)

Proof of Theorem caucvgsrlemoffcau

Dummy variables f g h are mutually distinct and distinct from all other variables.

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	. . . . . . . . . . . 12 |- ( ph -> F : N. --> R. )
3		caucvgsrlembnd.bnd	. . . . . . . . . . . 12 |- ( ph -> A. m e. N. A <R ( F ` m ) )
4		caucvgsrlembnd.offset	. . . . . . . . . . . 12 |- G = ( a e. N. |-> ( ( ( F ` a ) +R 1R ) +R ( A .R -1R ) ) )
5	2, 1, 3, 4	caucvgsrlemoffval 7797	. . . . . . . . . . 11 |- ( ( ph /\ n e. N. ) -> ( ( G ` n ) +R A ) = ( ( F ` n ) +R 1R ) )
6	5	adantr 276	. . . . . . . . . 10 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( G ` n ) +R A ) = ( ( F ` n ) +R 1R ) )
7	6	eqcomd 2183	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( F ` n ) +R 1R ) = ( ( G ` n ) +R A ) )
8	2	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> F : N. --> R. )
9		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> k e. N. )
10	8, 9	ffvelcdmd 5654	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( F ` k ) e. R. )
11		simplr 528	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> n e. N. )
12		recnnpr 7549	. . . . . . . . . . . 12 |- ( n e. N. -> <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. e. P. )
13		prsrcl 7785	. . . . . . . . . . . 12 |- ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. e. P. -> [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. )
14	11, 12, 13	3syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. )
15		1sr 7752	. . . . . . . . . . . 12 |- 1R e. R.
16	15	a1i 9	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> 1R e. R. )
17		addcomsrg 7756	. . . . . . . . . . . 12 |- ( ( f e. R. /\ g e. R. ) -> ( f +R g ) = ( g +R f ) )
18	17	adantl 277	. . . . . . . . . . 11 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ ( f e. R. /\ g e. R. ) ) -> ( f +R g ) = ( g +R f ) )
19		addasssrg 7757	. . . . . . . . . . . 12 |- ( ( f e. R. /\ g e. R. /\ h e. R. ) -> ( ( f +R g ) +R h ) = ( f +R ( g +R h ) ) )
20	19	adantl 277	. . . . . . . . . . 11 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ ( f e. R. /\ g e. R. /\ h e. R. ) ) -> ( ( f +R g ) +R h ) = ( f +R ( g +R h ) ) )
21	10, 14, 16, 18, 20	caov32d 6057	. . . . . . . . . 10 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R 1R ) = ( ( ( F ` k ) +R 1R ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
22	2, 1, 3, 4	caucvgsrlemoffval 7797	. . . . . . . . . . . 12 |- ( ( ph /\ k e. N. ) -> ( ( G ` k ) +R A ) = ( ( F ` k ) +R 1R ) )
23	22	adantlr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( G ` k ) +R A ) = ( ( F ` k ) +R 1R ) )
24	23	oveq1d 5892	. . . . . . . . . 10 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( G ` k ) +R A ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) = ( ( ( F ` k ) +R 1R ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
25	2, 1, 3, 4	caucvgsrlemofff 7798	. . . . . . . . . . . . 13 |- ( ph -> G : N. --> R. )
26	25	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> G : N. --> R. )
27	26, 9	ffvelcdmd 5654	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( G ` k ) e. R. )
28	3	caucvgsrlemasr 7791	. . . . . . . . . . . 12 |- ( ph -> A e. R. )
29	28	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> A e. R. )
30	27, 29, 14, 18, 20	caov32d 6057	. . . . . . . . . 10 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( G ` k ) +R A ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) = ( ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R A ) )
31	21, 24, 30	3eqtr2d 2216	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R 1R ) = ( ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R A ) )
32	7, 31	breq12d 4018	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( F ` n ) +R 1R ) <R ( ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R 1R ) <-> ( ( G ` n ) +R A ) <R ( ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R A ) ) )
33		ltasrg 7771	. . . . . . . . . 10 |- ( ( f e. R. /\ g e. R. /\ h e. R. ) -> ( f <R g <-> ( h +R f ) <R ( h +R g ) ) )
34	33	adantl 277	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ ( f e. R. /\ g e. R. /\ h e. R. ) ) -> ( f <R g <-> ( h +R f ) <R ( h +R g ) ) )
35	8, 11	ffvelcdmd 5654	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( F ` n ) e. R. )
36		addclsr 7754	. . . . . . . . . 10 |- ( ( ( F ` k ) e. R. /\ [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. ) -> ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) e. R. )
37	10, 14, 36	syl2anc 411	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) e. R. )
38	34, 35, 37, 16, 18	caovord2d 6046	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( 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 ` n ) +R 1R ) <R ( ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R 1R ) ) )
39	26, 11	ffvelcdmd 5654	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( G ` n ) e. R. )
40		addclsr 7754	. . . . . . . . . 10 |- ( ( ( G ` k ) e. R. /\ [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. ) -> ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) e. R. )
41	27, 14, 40	syl2anc 411	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) e. R. )
42	34, 39, 41, 29, 18	caovord2d 6046	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) <-> ( ( G ` n ) +R A ) <R ( ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R A ) ) )
43	32, 38, 42	3bitr4d 220	. . . . . . 7 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) <-> ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
44	23	eqcomd 2183	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( F ` k ) +R 1R ) = ( ( G ` k ) +R A ) )
45	35, 14, 16, 18, 20	caov32d 6057	. . . . . . . . . 10 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R 1R ) = ( ( ( F ` n ) +R 1R ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
46	6	oveq1d 5892	. . . . . . . . . 10 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( G ` n ) +R A ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) = ( ( ( F ` n ) +R 1R ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
47	39, 29, 14, 18, 20	caov32d 6057	. . . . . . . . . 10 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( G ` n ) +R A ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) = ( ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R A ) )
48	45, 46, 47	3eqtr2d 2216	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R 1R ) = ( ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R A ) )
49	44, 48	breq12d 4018	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( F ` k ) +R 1R ) <R ( ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R 1R ) <-> ( ( G ` k ) +R A ) <R ( ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R A ) ) )
50		addclsr 7754	. . . . . . . . . 10 |- ( ( ( F ` n ) e. R. /\ [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. ) -> ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) e. R. )
51	35, 14, 50	syl2anc 411	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) e. R. )
52	34, 10, 51, 16, 18	caovord2d 6046	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) <-> ( ( F ` k ) +R 1R ) <R ( ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R 1R ) ) )
53		addclsr 7754	. . . . . . . . . 10 |- ( ( ( G ` n ) e. R. /\ [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. ) -> ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) e. R. )
54	39, 14, 53	syl2anc 411	. . . . . . . . 9 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) e. R. )
55	34, 27, 54, 29, 18	caovord2d 6046	. . . . . . . 8 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) <-> ( ( G ` k ) +R A ) <R ( ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) +R A ) ) )
56	49, 52, 55	3bitr4d 220	. . . . . . 7 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) <-> ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
57	43, 56	anbi12d 473	. . . . . 6 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( 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 ) ) <-> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
58	57	biimpd 144	. . . . 5 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( ( ( 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 ) ) -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
59	58	imim2d 54	. . . 4 |- ( ( ( ph /\ n e. N. ) /\ 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 ) ) ) -> ( n <N k -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) ) )
60	59	ralimdva 2544	. . 3 |- ( ( ph /\ 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 ) ) ) -> A. k e. N. ( n <N k -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) ) )
61	60	ralimdva 2544	. 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 ) ) ) -> A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) ) )
62	1, 61	mpd 13	1 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   <.cop 3597   class class class wbr 4005   |-> cmpt 4066   -->wf 5214   ` cfv 5218  (class class class)co 5877   1oc1o 6412   [cec 6535   N.cnpi 7273   <N clti 7276   ~Q ceq 7280   *Qcrq 7285   <Q cltq 7286   P.cnp 7292   1Pc1p 7293   +P. cpp 7294   ~R cer 7297   R.cnr 7298   1Rc1r 7300   -1Rcm1r 7301   +R cplr 7302   .R cmr 7303   <R cltr 7304

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-i1p 7468  df-iplp 7469  df-imp 7470  df-iltp 7471  df-enr 7727  df-nr 7728  df-plr 7729  df-mr 7730  df-ltr 7731  df-0r 7732  df-1r 7733  df-m1r 7734

This theorem is referenced by:  caucvgsrlemoffres  7801