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Theorem prodeq1 25306
Description: Equality theorem for a composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
Assertion
Ref Expression
prodeq1  |-  ( A  =  B  ->  prod_ k  e.  A G C  =  prod_ k  e.  B G C )

Proof of Theorem prodeq1
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2289 . . 3  |-  ( A  =  B  ->  ( A  =  (/)  <->  B  =  (/) ) )
2 eqeq1 2289 . . . . . . 7  |-  ( A  =  B  ->  ( A  =  ( m ... n )  <->  B  =  ( m ... n
) ) )
32anbi1d 685 . . . . . 6  |-  ( A  =  B  ->  (
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) )  <-> 
( B  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) ) )
43rexbidv 2564 . . . . 5  |-  ( A  =  B  ->  ( E. n  e.  ( ZZ>=
`  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  C ) ) `  n ) )  <->  E. n  e.  ( ZZ>= `  m )
( B  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) ) )
54exbidv 1612 . . . 4  |-  ( A  =  B  ->  ( E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) )  <->  E. m E. n  e.  ( ZZ>= `  m )
( B  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) ) )
65abbidv 2397 . . 3  |-  ( A  =  B  ->  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) }  =  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( B  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) } )
71, 6ifbieq2d 3585 . 2  |-  ( A  =  B  ->  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 ) ) } )  =  if ( B  =  (/) ,  (GId
`  G ) ,  { x  |  E. m E. n  e.  (
ZZ>= `  m ) ( B  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  C ) ) `  n ) ) } ) )
8 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 ) ) } )
9 df-prod 25299 . 2  |-  prod_ k  e.  B G C  =  if ( B  =  (/) ,  (GId `  G
) ,  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( B  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) } )
107, 8, 93eqtr4g 2340 1  |-  ( A  =  B  ->  prod_ k  e.  A G C  =  prod_ k  e.  B G C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   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:  prodeq1d  25313  fsumprd  25329
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-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-un 3157  df-if 3566  df-prod 25299
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