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Theorem prod0 25305
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 25299 . 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 2283 . . 3  |-  (/)  =  (/)
3 iftrue 3571 . . 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 2303 1  |-  prod_ k  e.  (/) G B  =  (GId `  G )
Colors of variables: wff set class
Syntax hints:    /\ 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:  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-if 3566  df-prod 25299
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