Theorem caucvgprlemlim 6985

Description: Lemma for caucvgpr 6986. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
caucvgprlemlim	|- ( ph -> A. x e. Q. E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( L +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) )

Distinct variable groups:   A, j   j, F, u, l, k   n, F, k   j, k, ph, x   k, l, u, x, j   j, L, k

Allowed substitution hints:   ph( u, n, l)   A( x, u, k, n, l)   F( x)   L( x, u, n, l)

Proof of Theorem caucvgprlemlim

Step	Hyp	Ref	Expression
1		archrecnq 6967	. . . 4 |- ( x e. Q. -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x )
2	1	adantl 271	. . 3 |- ( ( ph /\ x e. Q. ) -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x )
3		caucvgpr.f	. . . . . . . . . 10 |- ( ph -> F : N. --> Q. )
4	3	ad5antr 480	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. Q. ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x ) /\ k e. N. ) /\ j <N k ) -> F : N. --> Q. )
5		caucvgpr.cau	. . . . . . . . . 10 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
6	5	ad5antr 480	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. Q. ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x ) /\ k e. N. ) /\ j <N k ) -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
7		caucvgpr.bnd	. . . . . . . . . 10 |- ( ph -> A. j e. N. A <Q ( F ` j ) )
8	7	ad5antr 480	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. Q. ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x ) /\ k e. N. ) /\ j <N k ) -> A. j e. N. A <Q ( F ` j ) )
9		caucvgpr.lim	. . . . . . . . 9 |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
10		simpr 108	. . . . . . . . . 10 |- ( ( ph /\ x e. Q. ) -> x e. Q. )
11	10	ad4antr 478	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. Q. ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x ) /\ k e. N. ) /\ j <N k ) -> x e. Q. )
12		simpr 108	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. Q. ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x ) /\ k e. N. ) /\ j <N k ) -> j <N k )
13		simpllr 501	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ x e. Q. ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x ) /\ k e. N. ) /\ j <N k ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x )
14	4, 6, 8, 9, 11, 12, 13	caucvgprlem1 6983	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ x e. Q. ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x ) /\ k e. N. ) /\ j <N k ) -> <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( L +P. <. { l | l <Q x } , { u | x <Q u } >. ) )
15	4, 6, 8, 9, 11, 12, 13	caucvgprlem2 6984	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ x e. Q. ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x ) /\ k e. N. ) /\ j <N k ) -> L <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. )
16	14, 15	jca 300	. . . . . . 7 |- ( ( ( ( ( ( ph /\ x e. Q. ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x ) /\ k e. N. ) /\ j <N k ) -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( L +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) )
17	16	ex 113	. . . . . 6 |- ( ( ( ( ( ph /\ x e. Q. ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x ) /\ k e. N. ) -> ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( L +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) )
18	17	ralrimiva 2439	. . . . 5 |- ( ( ( ( ph /\ x e. Q. ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x ) -> A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( L +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) )
19	18	ex 113	. . . 4 |- ( ( ( ph /\ x e. Q. ) /\ j e. N. ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x -> A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( L +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) ) )
20	19	reximdva 2468	. . 3 |- ( ( ph /\ x e. Q. ) -> ( E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x -> E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( L +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) ) )
21	2, 20	mpd 13	. 2 |- ( ( ph /\ x e. Q. ) -> E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( L +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) )
22	21	ralrimiva 2439	1 |- ( ph -> A. x e. Q. E. j e. N. A. k e. N. ( j <N k -> ( <. { l | l <Q ( F ` k ) } , { u | ( F ` k ) <Q u } >. <P ( L +P. <. { l | l <Q x } , { u | x <Q u } >. ) /\ L <P <. { l | l <Q ( ( F ` k ) +Q x ) } , { u | ( ( F ` k ) +Q x ) <Q u } >. ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   {crab 2357   <.cop 3419   class class class wbr 3805   -->wf 4948   ` cfv 4952  (class class class)co 5563   1oc1o 6078   [cec 6191   N.cnpi 6576   <N clti 6579   ~Q ceq 6583   Q.cnq 6584   +Q cplq 6586   *Qcrq 6588   <Q cltq 6589   +P. cpp 6597   <P cltp 6599

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-1o 6085  df-2o 6086  df-oadd 6089  df-omul 6090  df-er 6193  df-ec 6195  df-qs 6199  df-ni 6608  df-pli 6609  df-mi 6610  df-lti 6611  df-plpq 6648  df-mpq 6649  df-enq 6651  df-nqqs 6652  df-plqqs 6653  df-mqqs 6654  df-1nqqs 6655  df-rq 6656  df-ltnqqs 6657  df-enq0 6728  df-nq0 6729  df-0nq0 6730  df-plq0 6731  df-mq0 6732  df-inp 6770  df-iplp 6772  df-iltp 6774

This theorem is referenced by:  caucvgpr  6986