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Theorem prodeq1 25409
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 2302 . . 3  |-  ( A  =  B  ->  ( A  =  (/)  <->  B  =  (/) ) )
2 eqeq1 2302 . . . . . . 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 2577 . . . . 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 1616 . . . 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 2410 . . 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 3598 . 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 25402 . 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 25402 . 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 2353 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   (/)c0 3468   ifcif 3578    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062  GIdcgi 20870   prod_cprd 25401
This theorem is referenced by:  prodeq1d  25416  fsumprd  25432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-un 3170  df-if 3579  df-prod 25402
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