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Theorem nfprod 24723
Description: Bound-variable hypothesis builder for  prod_. If  x is (effectively) not free in  A,  G and  B, it is not free in  prod_ k  e.  A G B. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
nfprod.1  |-  F/_ x A
nfprod.2  |-  F/_ x G
nfprod.3  |-  F/_ x B
Assertion
Ref Expression
nfprod  |-  F/_ x prod_ k  e.  A G B
Distinct variable group:    x, k
Allowed substitution hints:    A( x, k)    B( x, k)    G( x, k)

Proof of Theorem nfprod
Dummy variables  m  n  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-prod 24711 . 2  |-  prod_ k  e.  A G B  =  if ( A  =  (/) ,  (GId `  G
) ,  { z  |  E. m E. n  e.  ( ZZ>= `  m ) ( A  =  ( m ... n )  /\  z  e.  (  seq  m ( G ,  ( k  e.  _V  |->  B ) ) `  n ) ) } )
2 nfprod.1 . . . 4  |-  F/_ x A
32nfeq1 2428 . . 3  |-  F/ x  A  =  (/)
4 nfcv 2419 . . . 4  |-  F/_ xGId
5 nfprod.2 . . . 4  |-  F/_ x G
64, 5nffv 5532 . . 3  |-  F/_ x
(GId `  G )
7 nfcv 2419 . . . . . 6  |-  F/_ x
( ZZ>= `  m )
82nfeq1 2428 . . . . . . 7  |-  F/ x  A  =  ( m ... n )
9 nfcv 2419 . . . . . . . . . 10  |-  F/_ x m
10 nfcv 2419 . . . . . . . . . . 11  |-  F/_ x _V
11 nfprod.3 . . . . . . . . . . 11  |-  F/_ x B
1210, 11nfmpt 4108 . . . . . . . . . 10  |-  F/_ x
( k  e.  _V  |->  B )
139, 5, 12nfseq 11056 . . . . . . . . 9  |-  F/_ x  seq  m ( G , 
( k  e.  _V  |->  B ) )
14 nfcv 2419 . . . . . . . . 9  |-  F/_ x n
1513, 14nffv 5532 . . . . . . . 8  |-  F/_ x
(  seq  m ( G ,  ( k  e.  _V  |->  B ) ) `
 n )
1615nfcri 2413 . . . . . . 7  |-  F/ x  z  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n )
178, 16nfan 1771 . . . . . 6  |-  F/ x
( A  =  ( m ... n )  /\  z  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  B ) ) `
 n ) )
187, 17nfrex 2598 . . . . 5  |-  F/ x E. n  e.  ( ZZ>=
`  m ) ( A  =  ( m ... n )  /\  z  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) )
1918nfex 1767 . . . 4  |-  F/ x E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  z  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  B ) ) `
 n ) )
2019nfab 2423 . . 3  |-  F/_ x { z  |  E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  z  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) ) }
213, 6, 20nfif 3589 . 2  |-  F/_ x if ( A  =  (/) ,  (GId `  G ) ,  { z  |  E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  z  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) ) } )
221, 21nfcxfr 2416 1  |-  F/_ x prod_ k  e.  A G B
Colors of variables: wff set class
Syntax hints:    /\ 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 24710
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-xp 4695  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 24711
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