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Theorem caucvgprlemlol 7732
Description: Lemma for caucvgpr 7744. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-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
caucvgprlemlol |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )
Distinct variable groups:   A, j   F, l, r, s   u, F   j, L, r, s   j, l, s   ph, j, r, s   u, j, r, s
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 caucvgprlemlol
Dummy variables f g h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 7427 . . . . 5 |- <Q C_ ( Q. X. Q. )
21brel 4712 . . . 4 |- ( s <Q r -> ( s e. Q. /\ r e. Q. ) )
32simpld 112 . . 3 |- ( s <Q r -> s e. Q. )
433ad2ant2 1021 . 2 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. Q. )
5 oveq1 5926 . . . . . . . 8 |- ( l = r -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
65breq1d 4040 . . . . . . 7 |- ( l = r -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
76rexbidv 2495 . . . . . 6 |- ( l = r -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
8 caucvgpr.lim . . . . . . . 8 |- 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 } >.
98fveq2i 5558 . . . . . . 7 |- ( 1st ` L ) = ( 1st ` <. { 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 } >. )
10 nqex 7425 . . . . . . . . 9 |- Q. e. _V
1110rabex 4174 . . . . . . . 8 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
1210rabex 4174 . . . . . . . 8 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
1311, 12op1st 6201 . . . . . . 7 |- ( 1st ` <. { 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 } >. ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
149, 13eqtri 2214 . . . . . 6 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
157, 14elrab2 2920 . . . . 5 |- ( r e. ( 1st ` L ) <-> ( r e. Q. /\ E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
1615simprbi 275 . . . 4 |- ( r e. ( 1st ` L ) -> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
17163ad2ant3 1022 . . 3 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
18 simpll2 1039 . . . . . . 7 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> s <Q r )
19 ltanqg 7462 . . . . . . . . 9 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
2019adantl 277 . . . . . . . 8 |- ( ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
214ad2antrr 488 . . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> s e. Q. )
222simprd 114 . . . . . . . . . 10 |- ( s <Q r -> r e. Q. )
23223ad2ant2 1021 . . . . . . . . 9 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> r e. Q. )
2423ad2antrr 488 . . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> r e. Q. )
25 simplr 528 . . . . . . . . 9 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> j e. N. )
26 nnnq 7484 . . . . . . . . 9 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
27 recclnq 7454 . . . . . . . . 9 |- ( [ <. j , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. )
2825, 26, 273syl 17 . . . . . . . 8 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. )
29 addcomnqg 7443 . . . . . . . . 9 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
3029adantl 277 . . . . . . . 8 |- ( ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
3120, 21, 24, 28, 30caovord2d 6090 . . . . . . 7 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> ( s <Q r <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) ) )
3218, 31mpbid 147 . . . . . 6 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
33 ltsonq 7460 . . . . . . 7 |- <Q Or Q.
3433, 1sotri 5062 . . . . . 6 |- ( ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
3532, 34sylancom 420 . . . . 5 |- ( ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) /\ ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
3635ex 115 . . . 4 |- ( ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) /\ j e. N. ) -> ( ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
3736reximdva 2596 . . 3 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> ( E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) -> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
3817, 37mpd 13 . 2 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
39 oveq1 5926 . . . . 5 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
4039breq1d 4040 . . . 4 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
4140rexbidv 2495 . . 3 |- ( l = s -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
4241, 14elrab2 2920 . 2 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
434, 38, 42sylanbrc 417 1 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   {crab 2476   <.cop 3622   class class class wbr 4030   -->wf 5251   ` cfv 5255  (class class class)co 5919   1stc1st 6193   1oc1o 6464   [cec 6587   N.cnpi 7334   <N clti 7337   ~Q ceq 7341   Q.cnq 7342   +Q cplq 7344   *Qcrq 7346   <Q cltq 7347
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621
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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415
This theorem is referenced by:  caucvgprlemrnd  7735
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