Theorem caucvgprlemlim 7580

Description: Lemma for caucvgpr 7581. 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 7562	. . . 4 |- ( x e. Q. -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x )
2	1	adantl 275	. . 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 488	. . . . . . . . 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 488	. . . . . . . . 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 488	. . . . . . . . 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 109	. . . . . . . . . 10 |- ( ( ph /\ x e. Q. ) -> x e. Q. )
11	10	ad4antr 486	. . . . . . . . 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 109	. . . . . . . . 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 524	. . . . . . . . 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 7578	. . . . . . . 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 7579	. . . . . . . 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 304	. . . . . . 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 114	. . . . . 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 2527	. . . . 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 114	. . . 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 2556	. . 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 2527	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 103   = wceq 1332   e. wcel 2125   {cab 2140   A.wral 2432   E.wrex 2433   {crab 2436   <.cop 3559   class class class wbr 3961   -->wf 5159   ` cfv 5163  (class class class)co 5814   1oc1o 6346   [cec 6467   N.cnpi 7171   <N clti 7174   ~Q ceq 7178   Q.cnq 7179   +Q cplq 7181   *Qcrq 7183   <Q cltq 7184   +P. cpp 7192   <P cltp 7194

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-eprel 4244  df-id 4248  df-po 4251  df-iso 4252  df-iord 4321  df-on 4323  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-recs 6242  df-irdg 6307  df-1o 6353  df-2o 6354  df-oadd 6357  df-omul 6358  df-er 6469  df-ec 6471  df-qs 6475  df-ni 7203  df-pli 7204  df-mi 7205  df-lti 7206  df-plpq 7243  df-mpq 7244  df-enq 7246  df-nqqs 7247  df-plqqs 7248  df-mqqs 7249  df-1nqqs 7250  df-rq 7251  df-ltnqqs 7252  df-enq0 7323  df-nq0 7324  df-0nq0 7325  df-plq0 7326  df-mq0 7327  df-inp 7365  df-iplp 7367  df-iltp 7369

This theorem is referenced by:  caucvgpr  7581