Theorem caucvgprlemlim 7643

Description: Lemma for caucvgpr 7644. 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 7625	. . . 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 493	. . . . . . . . 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 493	. . . . . . . . 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 493	. . . . . . . . 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 491	. . . . . . . . 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 529	. . . . . . . . 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 7641	. . . . . . . 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 7642	. . . . . . . 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 2543	. . . . 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 2572	. . 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 2543	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 1348   e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   1oc1o 6388   [cec 6511   N.cnpi 7234   <N clti 7237   ~Q ceq 7241   Q.cnq 7242   +Q cplq 7244   *Qcrq 7246   <Q cltq 7247   +P. cpp 7255   <P cltp 7257

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432

This theorem is referenced by:  caucvgpr  7644