Theorem caucvgprprlemlim 7823

Description: Lemma for caucvgprpr 7824. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.)

Hypotheses

Ref	Expression
caucvgprpr.f	|- ( ph -> F : N. --> P. )
caucvgprpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
caucvgprpr.bnd	|- ( ph -> A. m e. N. A <P ( F ` m ) )
caucvgprpr.lim	|- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.

Assertion

Ref	Expression
caucvgprprlemlim	|- ( ph -> A. x e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <P ( L +P. x ) /\ L <P ( ( F ` k ) +P. x ) ) ) )

Distinct variable groups:   A, m   m, F   A, r, j   u, F, r, l, k, n   ph, k, r   k, L   j, k, ph, x   k, l, u, p, q, r   j, r, x   q, l, r   u, p, q, r   m, r   k, n, u, l   j, l, u   n, r

Allowed substitution hints:   ph( u, m, n, q, p, l)   A( x, u, k, n, q, p, l)   F( x, j, q, p)   L( x, u, j, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemlim

Step	Hyp	Ref	Expression
1		archrecpr 7776	. . . 4 |- ( x e. P. -> E. j e. N. <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x )
2	1	adantl 277	. . 3 |- ( ( ph /\ x e. P. ) -> E. j e. N. <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x )
3		caucvgprpr.f	. . . . . . . . . 10 |- ( ph -> F : N. --> P. )
4	3	ad5antr 496	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. P. ) /\ j e. N. ) /\ <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x ) /\ k e. N. ) /\ j <N k ) -> F : N. --> P. )
5		caucvgprpr.cau	. . . . . . . . . 10 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
6	5	ad5antr 496	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. P. ) /\ j e. N. ) /\ <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x ) /\ k e. N. ) /\ j <N k ) -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
7		caucvgprpr.bnd	. . . . . . . . . 10 |- ( ph -> A. m e. N. A <P ( F ` m ) )
8	7	ad5antr 496	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. P. ) /\ j e. N. ) /\ <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x ) /\ k e. N. ) /\ j <N k ) -> A. m e. N. A <P ( F ` m ) )
9		caucvgprpr.lim	. . . . . . . . 9 |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
10		simpr 110	. . . . . . . . . 10 |- ( ( ph /\ x e. P. ) -> x e. P. )
11	10	ad4antr 494	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. P. ) /\ j e. N. ) /\ <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x ) /\ k e. N. ) /\ j <N k ) -> x e. P. )
12		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. P. ) /\ j e. N. ) /\ <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x ) /\ k e. N. ) /\ j <N k ) -> j <N k )
13		simpllr 534	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. P. ) /\ j e. N. ) /\ <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x ) /\ k e. N. ) /\ j <N k ) -> <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x )
14	4, 6, 8, 9, 11, 12, 13	caucvgprprlem1 7821	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ x e. P. ) /\ j e. N. ) /\ <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x ) /\ k e. N. ) /\ j <N k ) -> ( F ` k ) <P ( L +P. x ) )
15	4, 6, 8, 9, 11, 12, 13	caucvgprprlem2 7822	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ x e. P. ) /\ j e. N. ) /\ <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x ) /\ k e. N. ) /\ j <N k ) -> L <P ( ( F ` k ) +P. x ) )
16	14, 15	jca 306	. . . . . . 7 |- ( ( ( ( ( ( ph /\ x e. P. ) /\ j e. N. ) /\ <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x ) /\ k e. N. ) /\ j <N k ) -> ( ( F ` k ) <P ( L +P. x ) /\ L <P ( ( F ` k ) +P. x ) ) )
17	16	ex 115	. . . . . 6 |- ( ( ( ( ( ph /\ x e. P. ) /\ j e. N. ) /\ <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x ) /\ k e. N. ) -> ( j <N k -> ( ( F ` k ) <P ( L +P. x ) /\ L <P ( ( F ` k ) +P. x ) ) ) )
18	17	ralrimiva 2578	. . . . 5 |- ( ( ( ( ph /\ x e. P. ) /\ j e. N. ) /\ <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x ) -> A. k e. N. ( j <N k -> ( ( F ` k ) <P ( L +P. x ) /\ L <P ( ( F ` k ) +P. x ) ) ) )
19	18	ex 115	. . . 4 |- ( ( ( ph /\ x e. P. ) /\ j e. N. ) -> ( <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x -> A. k e. N. ( j <N k -> ( ( F ` k ) <P ( L +P. x ) /\ L <P ( ( F ` k ) +P. x ) ) ) ) )
20	19	reximdva 2607	. . 3 |- ( ( ph /\ x e. P. ) -> ( E. j e. N. <. { l | l <Q ( *Q ` [ <. j , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. j , 1o >. ] ~Q ) <Q u } >. <P x -> E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <P ( L +P. x ) /\ L <P ( ( F ` k ) +P. x ) ) ) ) )
21	2, 20	mpd 13	. 2 |- ( ( ph /\ x e. P. ) -> E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <P ( L +P. x ) /\ L <P ( ( F ` k ) +P. x ) ) ) )
22	21	ralrimiva 2578	1 |- ( ph -> A. x e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <P ( L +P. x ) /\ L <P ( ( F ` k ) +P. x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   {cab 2190   A.wral 2483   E.wrex 2484   {crab 2487   <.cop 3635   class class class wbr 4043   -->wf 5266   ` cfv 5270  (class class class)co 5943   1oc1o 6494   [cec 6617   N.cnpi 7384   <N clti 7387   ~Q ceq 7391   Q.cnq 7392   +Q cplq 7394   *Qcrq 7396   <Q cltq 7397   P.cnp 7403   +P. cpp 7405   <P cltp 7407

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-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-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-iplp 7580  df-iltp 7582

This theorem is referenced by:  caucvgprpr  7824