Theorem caucvgprlemlim 8038

Description: Lemma for caucvgpr 8039. 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 8020	. . . 4 |- ( x e. Q. -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) <Q x )
2	1	adantl 277	. . 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 500	. . . . . . . . 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 500	. . . . . . . . 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 500	. . . . . . . . 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 110	. . . . . . . . . 10 |- ( ( ph /\ x e. Q. ) -> x e. Q. )
11	10	ad4antr 498	. . . . . . . . 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 110	. . . . . . . . 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 540	. . . . . . . . 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 8036	. . . . . . . 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 8037	. . . . . . . 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 306	. . . . . . 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 115	. . . . . 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 2623	. . . . 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 115	. . . 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 2652	. . 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 2623	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 104   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   {crab 2532   <.cop 3708   class class class wbr 4125   -->wf 5368   ` cfv 5372  (class class class)co 6075   1oc1o 6670   [cec 6795   N.cnpi 7629   <N clti 7632   ~Q ceq 7636   Q.cnq 7637   +Q cplq 7639   *Qcrq 7641   <Q cltq 7642   +P. cpp 7650   <P cltp 7652

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iplp 7825  df-iltp 7827

This theorem is referenced by:  caucvgpr  8039