Theorem caucvgprlemrnd 6644

Description: Lemma for caucvgpr 6653. 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 6640	. . . . 5 |- ((ph /\ s e. ( 1st ` L)) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L)))
6	5	ex 108	. . . 4 |- (ph -> ( s e. ( 1st ` L) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L))))
7	1, 2, 3, 4	caucvgprlemlol 6641	. . . . . 6 |- ((ph /\ s <Q r /\ r e. ( 1st ` L)) -> s e. ( 1st ` L))
8	7	3expib 1106	. . . . 5 |- (ph -> (( s <Q r /\ r e. ( 1st ` L)) -> s e. ( 1st ` L)))
9	8	rexlimdvw 2430	. . . 4 |- (ph -> (E. r e. Q. ( s <Q r /\ r e. ( 1st ` L)) -> s e. ( 1st ` L)))
10	6, 9	impbid 120	. . 3 |- (ph -> ( s e. ( 1st ` L) <-> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L))))
11	10	ralrimivw 2387	. 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 6642	. . . . 5 |- ((ph /\ r e. ( 2nd ` L)) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L)))
13	12	ex 108	. . . 4 |- (ph -> ( r e. ( 2nd ` L) -> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L))))
14	1, 2, 3, 4	caucvgprlemupu 6643	. . . . . 6 |- ((ph /\ s <Q r /\ s e. ( 2nd ` L)) -> r e. ( 2nd ` L))
15	14	3expib 1106	. . . . 5 |- (ph -> (( s <Q r /\ s e. ( 2nd ` L)) -> r e. ( 2nd ` L)))
16	15	rexlimdvw 2430	. . . 4 |- (ph -> (E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L)) -> r e. ( 2nd ` L)))
17	13, 16	impbid 120	. . 3 |- (ph -> ( r e. ( 2nd ` L) <-> E. s e. Q. ( s <Q r /\ s e. ( 2nd ` L))))
18	17	ralrimivw 2387	. 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 290	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 97   <-> wb 98   = wceq 1242   e. wcel 1390  A.wral 2300  E.wrex 2301   {crab 2304   <.cop 3370   class class class wbr 3755   -->wf 4841   ` cfv 4845  (class class class)co 5455   1stc1st 5707   2ndc2nd 5708   1oc1o 5933  [cec 6040   N.cnpi 6256   <N clti 6259   ~Q ceq 6263   Q.cnq 6264   +Q cplq 6266   *Qcrq 6268   <Q cltq 6269

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337

This theorem is referenced by:  caucvgprlemcl  6647