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Theorem caucvgprlemlol 7446
Description: Lemma for caucvgpr 7458. 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 7141 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4561 . . . 4  |-  ( s 
<Q  r  ->  ( s  e.  Q.  /\  r  e.  Q. ) )
32simpld 111 . . 3  |-  ( s 
<Q  r  ->  s  e. 
Q. )
433ad2ant2 988 . 2  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  s  e.  Q. )
5 oveq1 5749 . . . . . . . 8  |-  ( l  =  r  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
65breq1d 3909 . . . . . . 7  |-  ( l  =  r  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
76rexbidv 2415 . . . . . 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 5392 . . . . . . 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 7139 . . . . . . . . 9  |-  Q.  e.  _V
1110rabex 4042 . . . . . . . 8  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
1210rabex 4042 . . . . . . . 8  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
1311, 12op1st 6012 . . . . . . 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 2138 . . . . . 6  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
157, 14elrab2 2816 . . . . 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 989 . . 3  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  E. j  e.  N.  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
18 simpll2 1006 . . . . . . 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 7176 . . . . . . . . 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 988 . . . . . . . . 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 504 . . . . . . . . 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 7198 . . . . . . . . 9  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
27 recclnq 7168 . . . . . . . . 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 7157 . . . . . . . . 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 5908 . . . . . . 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 7174 . . . . . . 7  |-  <Q  Or  Q.
3433, 1sotri 4904 . . . . . 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 2511 . . 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 5749 . . . . 5  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
4039breq1d 3909 . . . 4  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
4140rexbidv 2415 . . 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 2816 . 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 947    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   {crab 2397   <.cop 3500   class class class wbr 3899   -->wf 5089   ` cfv 5093  (class class class)co 5742   1stc1st 6004   1oc1o 6274   [cec 6395   N.cnpi 7048    <N clti 7051    ~Q ceq 7055   Q.cnq 7056    +Q cplq 7058   *Qcrq 7060    <Q cltq 7061
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129
This theorem is referenced by:  caucvgprlemrnd  7449
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