Theorem caucvgprlemlim 7829

Description: Lemma for caucvgpr 7830. 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 7811	. . . 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 496	. . . . . . . . 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 496	. . . . . . . . 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 496	. . . . . . . . 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 494	. . . . . . . . 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 534	. . . . . . . . 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 7827	. . . . . . . 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 7828	. . . . . . . 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 2581	. . . . 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 2610	. . 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 2581	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 1373   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   {crab 2490   <.cop 3646   class class class wbr 4059   -->wf 5286   ` cfv 5290  (class class class)co 5967   1oc1o 6518   [cec 6641   N.cnpi 7420   <N clti 7423   ~Q ceq 7427   Q.cnq 7428   +Q cplq 7430   *Qcrq 7432   <Q cltq 7433   +P. cpp 7441   <P cltp 7443

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-2o 6526  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-enq0 7572  df-nq0 7573  df-0nq0 7574  df-plq0 7575  df-mq0 7576  df-inp 7614  df-iplp 7616  df-iltp 7618

This theorem is referenced by:  caucvgpr  7830