Theorem caucvgprprlemrnd 8016

Description: Lemma for caucvgprpr 8027. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.)

Hypotheses

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

Assertion

Ref	Expression
caucvgprprlemrnd	|- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) ) /\ A. t e. Q. ( t e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) ) ) )

Distinct variable groups:   A, m   m, F   F, l, t   u, F, t, r, s   L, s, t   p, l, q, r, s, t   u, p, q, r, s   ph, r, s, t

Allowed substitution hints:   ph( u, k, m, n, q, p, l)   A( u, t, k, n, s, r, q, p, l)   F( k, n, q, p)   L( u, k, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemrnd

Step	Hyp	Ref	Expression
1		caucvgprpr.f	. . . . . 6 |- ( ph -> F : N. --> P. )
2		caucvgprpr.cau	. . . . . 6 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
3		caucvgprpr.bnd	. . . . . 6 |- ( ph -> A. m e. N. A <P ( F ` m ) )
4		caucvgprpr.lim	. . . . . 6 |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
5	1, 2, 3, 4	caucvgprprlemopl 8012	. . . . 5 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) )
6	5	ex 115	. . . 4 |- ( ph -> ( s e. ( 1st ` L ) -> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) ) )
7	1, 2, 3, 4	caucvgprprlemlol 8013	. . . . . 6 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )
8	7	3expib 1233	. . . . 5 |- ( ph -> ( ( s <Q t /\ t e. ( 1st ` L ) ) -> s e. ( 1st ` L ) ) )
9	8	rexlimdvw 2664	. . . 4 |- ( ph -> ( E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) -> s e. ( 1st ` L ) ) )
10	6, 9	impbid 129	. . 3 |- ( ph -> ( s e. ( 1st ` L ) <-> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) ) )
11	10	ralrimivw 2616	. 2 |- ( ph -> A. s e. Q. ( s e. ( 1st ` L ) <-> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) ) )
12	1, 2, 3, 4	caucvgprprlemopu 8014	. . . . 5 |- ( ( ph /\ t e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) )
13	12	ex 115	. . . 4 |- ( ph -> ( t e. ( 2nd ` L ) -> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) ) )
14	1, 2, 3, 4	caucvgprprlemupu 8015	. . . . . 6 |- ( ( ph /\ s <Q t /\ s e. ( 2nd ` L ) ) -> t e. ( 2nd ` L ) )
15	14	3expib 1233	. . . . 5 |- ( ph -> ( ( s <Q t /\ s e. ( 2nd ` L ) ) -> t e. ( 2nd ` L ) ) )
16	15	rexlimdvw 2664	. . . 4 |- ( ph -> ( E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) -> t e. ( 2nd ` L ) ) )
17	13, 16	impbid 129	. . 3 |- ( ph -> ( t e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) ) )
18	17	ralrimivw 2616	. 2 |- ( ph -> A. t e. Q. ( t e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) ) )
19	11, 18	jca 306	1 |- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. t e. Q. ( s <Q t /\ t e. ( 1st ` L ) ) ) /\ A. t e. Q. ( t e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   {cab 2218   A.wral 2520   E.wrex 2521   {crab 2524   <.cop 3692   class class class wbr 4109   -->wf 5348   ` cfv 5352  (class class class)co 6050   1stc1st 6332   2ndc2nd 6333   1oc1o 6640   [cec 6765   N.cnpi 7587   <N clti 7590   ~Q ceq 7594   Q.cnq 7595   +Q cplq 7597   *Qcrq 7599   <Q cltq 7600   P.cnp 7606   +P. cpp 7608   <P cltp 7610

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iplp 7783  df-iltp 7785

This theorem is referenced by:  caucvgprprlemcl  8019