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Theorem caucvgprlemlol 7478
Description: Lemma for caucvgpr 7490. 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 7173 . . . . 5 |- <Q C_ ( Q. X. Q. )
21brel 4591 . . . 4 |- ( s <Q r -> ( s e. Q. /\ r e. Q. ) )
32simpld 111 . . 3 |- ( s <Q r -> s e. Q. )
433ad2ant2 1003 . 2 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. Q. )
5 oveq1 5781 . . . . . . . 8 |- ( l = r -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
65breq1d 3939 . . . . . . 7 |- ( l = r -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
76rexbidv 2438 . . . . . 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 5424 . . . . . . 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 7171 . . . . . . . . 9 |- Q. e. _V
1110rabex 4072 . . . . . . . 8 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
1210rabex 4072 . . . . . . . 8 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
1311, 12op1st 6044 . . . . . . 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 2160 . . . . . 6 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
157, 14elrab2 2843 . . . . 5 |- ( r e. ( 1st ` L ) <-> ( r e. Q. /\ E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
1615simprbi 273 . . . 4 |- ( r e. ( 1st ` L ) -> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
17163ad2ant3 1004 . . 3 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
18 simpll2 1021 . . . . . . 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 7208 . . . . . . . . 9 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
2019adantl 275 . . . . . . . 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 479 . . . . . . . 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 113 . . . . . . . . . 10 |- ( s <Q r -> r e. Q. )
23223ad2ant2 1003 . . . . . . . . 9 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> r e. Q. )
2423ad2antrr 479 . . . . . . . 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 519 . . . . . . . . 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 7230 . . . . . . . . 9 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
27 recclnq 7200 . . . . . . . . 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 7189 . . . . . . . . 9 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
3029adantl 275 . . . . . . . 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 5940 . . . . . . 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 146 . . . . . 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 7206 . . . . . . 7 |- <Q Or Q.
3433, 1sotri 4934 . . . . . 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 416 . . . . 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 114 . . . 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 2534 . . 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 5781 . . . . 5 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
4039breq1d 3939 . . . 4 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
4140rexbidv 2438 . . 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 2843 . 2 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
434, 38, 42sylanbrc 413 1 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. ( 1st ` L ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   1stc1st 6036   1oc1o 6306   [cec 6427   N.cnpi 7080   <N clti 7083   ~Q ceq 7087   Q.cnq 7088   +Q cplq 7090   *Qcrq 7092   <Q cltq 7093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161
This theorem is referenced by:  caucvgprlemrnd  7481
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