Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cbvprodi Unicode version

Theorem cbvprodi 25312
Description: Change bound variable in a finite composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
Hypotheses
Ref Expression
cbvprod.1  |-  F/_ k B
cbvprod.2  |-  F/_ j C
cbvprod.3  |-  ( j  =  k  ->  B  =  C )
Assertion
Ref Expression
cbvprodi  |-  prod_ j  e.  A G B  = 
prod_ k  e.  A G C

Proof of Theorem cbvprodi
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvprod.1 . . . . . . . . . . 11  |-  F/_ k B
2 cbvprod.2 . . . . . . . . . . 11  |-  F/_ j C
3 cbvprod.3 . . . . . . . . . . 11  |-  ( j  =  k  ->  B  =  C )
41, 2, 3cbvmpt 4110 . . . . . . . . . 10  |-  ( j  e.  _V  |->  B )  =  ( k  e. 
_V  |->  C )
5 seqeq3 11051 . . . . . . . . . 10  |-  ( ( j  e.  _V  |->  B )  =  ( k  e.  _V  |->  C )  ->  seq  m ( G ,  ( j  e.  _V  |->  B ) )  =  seq  m ( G ,  ( k  e.  _V  |->  C ) ) )
64, 5ax-mp 8 . . . . . . . . 9  |-  seq  m
( G ,  ( j  e.  _V  |->  B ) )  =  seq  m ( G , 
( k  e.  _V  |->  C ) )
76fveq1i 5526 . . . . . . . 8  |-  (  seq  m ( G , 
( j  e.  _V  |->  B ) ) `  n )  =  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n )
87eleq2i 2347 . . . . . . 7  |-  ( x  e.  (  seq  m
( G ,  ( j  e.  _V  |->  B ) ) `  n
)  <->  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  C ) ) `  n ) )
98anbi2i 675 . . . . . 6  |-  ( ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( j  e.  _V  |->  B ) ) `  n ) )  <->  ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e.  _V  |->  C ) ) `  n ) ) )
109rexbii 2568 . . . . 5  |-  ( E. n  e.  ( ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( j  e.  _V  |->  B ) ) `  n ) )  <->  E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  C ) ) `  n ) ) )
1110exbii 1569 . . . 4  |-  ( E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( j  e. 
_V  |->  B ) ) `
 n ) )  <->  E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) )
1211abbii 2395 . . 3  |-  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( j  e. 
_V  |->  B ) ) `
 n ) ) }  =  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) }
13 ifeq2 3570 . . 3  |-  ( { x  |  E. m E. n  e.  ( ZZ>=
`  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( j  e.  _V  |->  B ) ) `  n ) ) }  =  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) }  ->  if ( A  =  (/) ,  (GId
`  G ) ,  { x  |  E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( j  e.  _V  |->  B ) ) `  n ) ) } )  =  if ( A  =  (/) ,  (GId
`  G ) ,  { x  |  E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  C ) ) `  n ) ) } ) )
1412, 13ax-mp 8 . 2  |-  if ( A  =  (/) ,  (GId
`  G ) ,  { x  |  E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( j  e.  _V  |->  B ) ) `  n ) ) } )  =  if ( A  =  (/) ,  (GId
`  G ) ,  { x  |  E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  C ) ) `  n ) ) } )
15 df-prod 25299 . 2  |-  prod_ j  e.  A G B  =  if ( A  =  (/) ,  (GId `  G
) ,  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( j  e. 
_V  |->  B ) ) `
 n ) ) } )
16 df-prod 25299 . 2  |-  prod_ k  e.  A G C  =  if ( A  =  (/) ,  (GId `  G
) ,  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) } )
1714, 15, 163eqtr4i 2313 1  |-  prod_ j  e.  A G B  = 
prod_ k  e.  A G C
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   F/_wnfc 2406   E.wrex 2544   _Vcvv 2788   (/)c0 3455   ifcif 3565    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046  GIdcgi 20854   prod_cprd 25298
This theorem is referenced by:  prodeqfv  25318  fprodserf  25321  fprod1s  25325  fprodp1s  25327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-seq 11047  df-prod 25299
  Copyright terms: Public domain W3C validator