Theorem caucvgprprlemlim 7771

Description: Lemma for caucvgprpr 7772. 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 7724	. . . 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 7769	. . . . . . . 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 7770	. . . . . . . 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 2567	. . . . 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 2596	. . 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 2567	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 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   {crab 2476   <.cop 3621   class class class wbr 4029   -->wf 5250   ` cfv 5254  (class class class)co 5918   1oc1o 6462   [cec 6585   N.cnpi 7332   <N clti 7335   ~Q ceq 7339   Q.cnq 7340   +Q cplq 7342   *Qcrq 7344   <Q cltq 7345   P.cnp 7351   +P. cpp 7353   <P cltp 7355

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530

This theorem is referenced by:  caucvgprpr  7772