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Theorem caucvgprlemlol 7660
Description: Lemma for caucvgpr 7672. 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 7355 . . . . 5 |- <Q C_ ( Q. X. Q. )
21brel 4675 . . . 4 |- ( s <Q r -> ( s e. Q. /\ r e. Q. ) )
32simpld 112 . . 3 |- ( s <Q r -> s e. Q. )
433ad2ant2 1019 . 2 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> s e. Q. )
5 oveq1 5876 . . . . . . . 8 |- ( l = r -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
65breq1d 4010 . . . . . . 7 |- ( l = r -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
76rexbidv 2478 . . . . . 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 5514 . . . . . . 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 7353 . . . . . . . . 9 |- Q. e. _V
1110rabex 4144 . . . . . . . 8 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
1210rabex 4144 . . . . . . . 8 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
1311, 12op1st 6141 . . . . . . 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 2198 . . . . . 6 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
157, 14elrab2 2896 . . . . 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 1020 . . 3 |- ( ( ph /\ s <Q r /\ r e. ( 1st ` L ) ) -> E. j e. N. ( r +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
18 simpll2 1037 . . . . . . 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 7390 . . . . . . . . 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 1019 . . . . . . . . 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 7412 . . . . . . . . 9 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
27 recclnq 7382 . . . . . . . . 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 7371 . . . . . . . . 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 6038 . . . . . . 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 7388 . . . . . . 7 |- <Q Or Q.
3433, 1sotri 5020 . . . . . 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 2579 . . 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 5876 . . . . 5 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
4039breq1d 4010 . . . 4 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
4140rexbidv 2478 . . 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 2896 . 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 978   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3594   class class class wbr 4000   -->wf 5208   ` cfv 5212  (class class class)co 5869   1stc1st 6133   1oc1o 6404   [cec 6527   N.cnpi 7262   <N clti 7265   ~Q ceq 7269   Q.cnq 7270   +Q cplq 7272   *Qcrq 7274   <Q cltq 7275
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343
This theorem is referenced by:  caucvgprlemrnd  7663
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