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Theorem caucvgprlemlol 8001
Description: Lemma for caucvgpr 8013. 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 7696 . . . . 5 |- <Q C_ ( Q. X. Q. )
21brel 4807 . . . 4 |- ( s <Q r -> ( s e. Q. /\ r e. Q. ) )
32simpld 112 . . 3 |- ( s <Q r -> s e. Q. )
433ad2ant2 1046 . 2 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. Q. )
5 oveq1 6065 . . . . . . . 8 |- ( l = r -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
65breq1d 4124 . . . . . . 7 |- ( l = r -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
76rexbidv 2545 . . . . . 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 5678 . . . . . . 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 7694 . . . . . . . . 9 |- Q. e. _V
1110rabex 4261 . . . . . . . 8 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
1210rabex 4261 . . . . . . . 8 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
1311, 12op1st 6353 . . . . . . 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 2255 . . . . . 6 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
157, 14elrab2 2979 . . . . 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 1047 . . 3 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
18 simpll2 1064 . . . . . . 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 7731 . . . . . . . . 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 1046 . . . . . . . . 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 529 . . . . . . . . 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 7753 . . . . . . . . 9 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
27 recclnq 7723 . . . . . . . . 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 7712 . . . . . . . . 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 6232 . . . . . . 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 7729 . . . . . . 7 |- <Q Or Q.
3433, 1sotri 5163 . . . . . 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 2646 . . 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 6065 . . . . 5 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
4039breq1d 4124 . . . 4 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
4140rexbidv 2545 . . 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 2979 . 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 1005   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   {crab 2526   <.cop 3697   class class class wbr 4114   -->wf 5353   ` cfv 5357  (class class class)co 6058   1stc1st 6345   1oc1o 6653   [cec 6778   N.cnpi 7603   <N clti 7606   ~Q ceq 7610   Q.cnq 7611   +Q cplq 7613   *Qcrq 7615   <Q cltq 7616
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684
This theorem is referenced by:  caucvgprlemrnd  8004
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