Theorem caucvgprlemrnd 7740

Description: Lemma for caucvgpr 7749. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-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
caucvgprlemrnd	|- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) )

Distinct variable groups:   A, j   L, r, s   ph, r, s   F, l, r, s   u, F, s   j, L, r   j, l, s   ph, j   u, j, r

Allowed substitution hints:   ph( u, k, n, l)   A( u, k, n, s, r, l)   F( j, k, n)   L( u, k, n, l)

Proof of Theorem caucvgprlemrnd

Step	Hyp	Ref	Expression
1		caucvgpr.f	. . . . . 6 |- ( ph -> F : N. --> Q. )
2		caucvgpr.cau	. . . . . 6 |- ( 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 ) ) ) ) )
3		caucvgpr.bnd	. . . . . 6 |- ( ph -> A. j e. N. A <Q ( F ` j ) )
4		caucvgpr.lim	. . . . . 6 |- 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 } >.
5	1, 2, 3, 4	caucvgprlemopl 7736	. . . . 5 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) )
6	5	ex 115	. . . 4 |- ( ph -> ( s e. ( 1st ` L ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) )
7	1, 2, 3, 4	caucvgprlemlol 7737	. . . . . 6 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )
8	7	3expib 1208	. . . . 5 |- ( ph -> ( ( s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) ) )
9	8	rexlimdvw 2618	. . . 4 |- ( ph -> ( E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) ) )
10	6, 9	impbid 129	. . 3 |- ( ph -> ( s e. ( 1st ` L ) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) )
11	10	ralrimivw 2571	. 2 |- ( ph -> A. s e. Q. ( s e. ( 1st ` L ) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) )
12	1, 2, 3, 4	caucvgprlemopu 7738	. . . . 5 |- ( ( ph /\ r e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) )
13	12	ex 115	. . . 4 |- ( ph -> ( r e. ( 2nd ` L ) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) )
14	1, 2, 3, 4	caucvgprlemupu 7739	. . . . . 6 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> r e. ( 2nd ` L ) )
15	14	3expib 1208	. . . . 5 |- ( ph -> ( ( s <Q r /\ s e. ( 2nd ` L ) ) -> r e. ( 2nd ` L ) ) )
16	15	rexlimdvw 2618	. . . 4 |- ( ph -> ( E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) -> r e. ( 2nd ` L ) ) )
17	13, 16	impbid 129	. . 3 |- ( ph -> ( r e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) )
18	17	ralrimivw 2571	. 2 |- ( ph -> A. r e. Q. ( r e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) )
19	11, 18	jca 306	1 |- ( ph -> ( A. s e. Q. ( s e. ( 1st ` L ) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` L ) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   {crab 2479   <.cop 3625   class class class wbr 4033   -->wf 5254   ` cfv 5258  (class class class)co 5922   1stc1st 6196   2ndc2nd 6197   1oc1o 6467   [cec 6590   N.cnpi 7339   <N clti 7342   ~Q ceq 7346   Q.cnq 7347   +Q cplq 7349   *Qcrq 7351   <Q cltq 7352

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420

This theorem is referenced by:  caucvgprlemcl  7743