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Theorem prod0 25408
Description: The value of  prod_ k  e.  (/) G B. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
Assertion
Ref Expression
prod0  |-  prod_ k  e.  (/) G B  =  (GId `  G )

Proof of Theorem prod0
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-prod 25402 . 2  |-  prod_ k  e.  (/) G B  =  if ( (/)  =  (/) ,  (GId `  G ) ,  { x  |  E. m E. n  e.  (
ZZ>= `  m ) (
(/)  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) ) } )
2 eqid 2296 . . 3  |-  (/)  =  (/)
3 iftrue 3584 . . 3  |-  ( (/)  =  (/)  ->  if ( (/)  =  (/) ,  (GId `  G ) ,  {
x  |  E. m E. n  e.  ( ZZ>=
`  m ) (
(/)  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) ) } )  =  (GId `  G ) )
42, 3ax-mp 8 . 2  |-  if (
(/)  =  (/) ,  (GId
`  G ) ,  { x  |  E. m E. n  e.  (
ZZ>= `  m ) (
(/)  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) ) } )  =  (GId `  G )
51, 4eqtri 2316 1  |-  prod_ k  e.  (/) G B  =  (GId `  G )
Colors of variables: wff set class
Syntax hints:    /\ 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:  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-if 3579  df-prod 25402
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