Theorem caucvgprlemrnd 7647

Description: Lemma for caucvgpr 7656. 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 7643	. . . . 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 7644	. . . . . 6 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )
8	7	3expib 1206	. . . . 5 |- ( ph -> ( ( s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) ) )
9	8	rexlimdvw 2596	. . . 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 2549	. 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 7645	. . . . 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 7646	. . . . . 6 |- ( ( ph /\ s <Q r /\ s e. ( 2nd ` L ) ) -> r e. ( 2nd ` L ) )
15	14	3expib 1206	. . . . 5 |- ( ph -> ( ( s <Q r /\ s e. ( 2nd ` L ) ) -> r e. ( 2nd ` L ) ) )
16	15	rexlimdvw 2596	. . . 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 2549	. 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 1353   e. wcel 2146   A.wral 2453   E.wrex 2454   {crab 2457   <.cop 3592   class class class wbr 3998   -->wf 5204   ` cfv 5208  (class class class)co 5865   1stc1st 6129   2ndc2nd 6130   1oc1o 6400   [cec 6523   N.cnpi 7246   <N clti 7249   ~Q ceq 7253   Q.cnq 7254   +Q cplq 7256   *Qcrq 7258   <Q cltq 7259

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327

This theorem is referenced by:  caucvgprlemcl  7650