Theorem caucvgprprlemrnd 7911

Description: Lemma for caucvgprpr 7922. 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 7907	. . . . 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 7908	. . . . . 6 |- ( ( ph /\ s <Q t /\ t e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )
8	7	3expib 1230	. . . . 5 |- ( ph -> ( ( s <Q t /\ t e. ( 1st ` L ) ) -> s e. ( 1st ` L ) ) )
9	8	rexlimdvw 2652	. . . 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 2604	. 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 7909	. . . . 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 7910	. . . . . 6 |- ( ( ph /\ s <Q t /\ s e. ( 2nd ` L ) ) -> t e. ( 2nd ` L ) )
15	14	3expib 1230	. . . . 5 |- ( ph -> ( ( s <Q t /\ s e. ( 2nd ` L ) ) -> t e. ( 2nd ` L ) ) )
16	15	rexlimdvw 2652	. . . 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 2604	. 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 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   {crab 2512   <.cop 3670   class class class wbr 4086   -->wf 5320   ` cfv 5324  (class class class)co 6013   1stc1st 6296   2ndc2nd 6297   1oc1o 6570   [cec 6695   N.cnpi 7482   <N clti 7485   ~Q ceq 7489   Q.cnq 7490   +Q cplq 7492   *Qcrq 7494   <Q cltq 7495   P.cnp 7501   +P. cpp 7503   <P cltp 7505

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-iplp 7678  df-iltp 7680

This theorem is referenced by:  caucvgprprlemcl  7914