Theorem caucvgprprlemlim 7519

Description: Lemma for caucvgprpr 7520. 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 7472	. . . 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 275	. . 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 487	. . . . . . . . 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 487	. . . . . . . . 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 487	. . . . . . . . 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 109	. . . . . . . . . 10 |- ( ( ph /\ x e. P. ) -> x e. P. )
11	10	ad4antr 485	. . . . . . . . 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 109	. . . . . . . . 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 523	. . . . . . . . 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 7517	. . . . . . . 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 7518	. . . . . . . 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 304	. . . . . . 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 114	. . . . . 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 2505	. . . . 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 114	. . . 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 2534	. . 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 2505	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 103   = wceq 1331   e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   1oc1o 6306   [cec 6427   N.cnpi 7080   <N clti 7083   ~Q ceq 7087   Q.cnq 7088   +Q cplq 7090   *Qcrq 7092   <Q cltq 7093   P.cnp 7099   +P. cpp 7101   <P cltp 7103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276  df-iltp 7278

This theorem is referenced by:  caucvgprpr  7520