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Theorem caucvgprprlemcbv 6766
Description: Lemma for caucvgprpr 6791. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)
Hypotheses
Ref Expression
caucvgprpr.f  |-  ( ph  ->  F : N. --> P. )
caucvgprpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
Assertion
Ref Expression
caucvgprprlemcbv  |-  ( ph  ->  A. a  e.  N.  A. b  e.  N.  (
a  <N  b  ->  (
( F `  a
)  <P  ( ( F `
 b )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  b
)  <P  ( ( F `
 a )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
Distinct variable groups:    F, a, b, k    n, F, a, k    a, l, b, k    u, a, b, k    n, l    u, n
Allowed substitution hints:    ph( u, k, n, a, b, l)    F( u, l)

Proof of Theorem caucvgprprlemcbv
StepHypRef Expression
1 caucvgprpr.cau . 2  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
2 breq1 3764 . . . 4  |-  ( n  =  a  ->  (
n  <N  k  <->  a  <N  k ) )
3 fveq2 5165 . . . . . 6  |-  ( n  =  a  ->  ( F `  n )  =  ( F `  a ) )
4 opeq1 3546 . . . . . . . . . . . 12  |-  ( n  =  a  ->  <. n ,  1o >.  =  <. a ,  1o >. )
54eceq1d 6129 . . . . . . . . . . 11  |-  ( n  =  a  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
65fveq2d 5169 . . . . . . . . . 10  |-  ( n  =  a  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
76breq2d 3773 . . . . . . . . 9  |-  ( n  =  a  ->  (
l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
87abbidv 2155 . . . . . . . 8  |-  ( n  =  a  ->  { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  ) } )
96breq1d 3771 . . . . . . . . 9  |-  ( n  =  a  ->  (
( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u  <->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u ) )
109abbidv 2155 . . . . . . . 8  |-  ( n  =  a  ->  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u }  =  {
u  |  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )  <Q  u } )
118, 10opeq12d 3554 . . . . . . 7  |-  ( n  =  a  ->  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  =  <. { l  |  l  <Q 
( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
1211oveq2d 5515 . . . . . 6  |-  ( n  =  a  ->  (
( F `  k
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  =  ( ( F `  k
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. ) )
133, 12breq12d 3774 . . . . 5  |-  ( n  =  a  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  <->  ( F `  a ) 
<P  ( ( F `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
) )
143, 11oveq12d 5517 . . . . . 6  |-  ( n  =  a  ->  (
( F `  n
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  =  ( ( F `  a
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. ) )
1514breq2d 3773 . . . . 5  |-  ( n  =  a  ->  (
( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  <->  ( F `  k ) 
<P  ( ( F `  a )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
) )
1613, 15anbi12d 442 . . . 4  |-  ( n  =  a  ->  (
( ( F `  n )  <P  (
( F `  k
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k )  <P  ( ( F `  n )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
)  <->  ( ( F `
 a )  <P 
( ( F `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 a )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
172, 16imbi12d 223 . . 3  |-  ( n  =  a  ->  (
( n  <N  k  ->  ( ( F `  n )  <P  (
( F `  k
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k )  <P  ( ( F `  n )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) )  <->  ( a  <N  k  ->  ( ( F `  a )  <P  ( ( F `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 a )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) ) )
18 breq2 3765 . . . 4  |-  ( k  =  b  ->  (
a  <N  k  <->  a  <N  b ) )
19 fveq2 5165 . . . . . . 7  |-  ( k  =  b  ->  ( F `  k )  =  ( F `  b ) )
2019oveq1d 5514 . . . . . 6  |-  ( k  =  b  ->  (
( F `  k
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  =  ( ( F `  b
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. ) )
2120breq2d 3773 . . . . 5  |-  ( k  =  b  ->  (
( F `  a
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  <->  ( F `  a ) 
<P  ( ( F `  b )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
) )
2219breq1d 3771 . . . . 5  |-  ( k  =  b  ->  (
( F `  k
)  <P  ( ( F `
 a )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  <->  ( F `  b ) 
<P  ( ( F `  a )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
) )
2321, 22anbi12d 442 . . . 4  |-  ( k  =  b  ->  (
( ( F `  a )  <P  (
( F `  k
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k )  <P  ( ( F `  a )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
)  <->  ( ( F `
 a )  <P 
( ( F `  b )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  b
)  <P  ( ( F `
 a )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
2418, 23imbi12d 223 . . 3  |-  ( k  =  b  ->  (
( a  <N  k  ->  ( ( F `  a )  <P  (
( F `  k
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k )  <P  ( ( F `  a )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
) )  <->  ( a  <N  b  ->  ( ( F `  a )  <P  ( ( F `  b )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  b
)  <P  ( ( F `
 a )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) ) )
2517, 24cbvral2v 2538 . 2  |-  ( A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  ( ( F `
 n )  <P 
( ( F `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) )  <->  A. a  e.  N.  A. b  e. 
N.  ( a  <N 
b  ->  ( ( F `  a )  <P  ( ( F `  b )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  b
)  <P  ( ( F `
 a )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
261, 25sylib 127 1  |-  ( ph  ->  A. a  e.  N.  A. b  e.  N.  (
a  <N  b  ->  (
( F `  a
)  <P  ( ( F `
 b )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  b
)  <P  ( ( F `
 a )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97   {cab 2026   A.wral 2303   <.cop 3375   class class class wbr 3761   -->wf 4885   ` cfv 4889  (class class class)co 5499   1oc1o 5981   [cec 6091   N.cnpi 6351    <N clti 6354    ~Q ceq 6358   *Qcrq 6363    <Q cltq 6364   P.cnp 6370    +P. cpp 6372    <P cltp 6374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-br 3762  df-opab 3816  df-xp 4338  df-cnv 4340  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-iota 4854  df-fv 4897  df-ov 5502  df-ec 6095
This theorem is referenced by:  caucvgprprlemval  6767
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