Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nfprod Unicode version

Theorem nfprod 25414
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 25402 . 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 2441 . . 3  |-  F/ x  A  =  (/)
4 nfcv 2432 . . . 4  |-  F/_ xGId
5 nfprod.2 . . . 4  |-  F/_ x G
64, 5nffv 5548 . . 3  |-  F/_ x
(GId `  G )
7 nfcv 2432 . . . . . 6  |-  F/_ x
( ZZ>= `  m )
82nfeq1 2441 . . . . . . 7  |-  F/ x  A  =  ( m ... n )
9 nfcv 2432 . . . . . . . . . 10  |-  F/_ x m
10 nfcv 2432 . . . . . . . . . . 11  |-  F/_ x _V
11 nfprod.3 . . . . . . . . . . 11  |-  F/_ x B
1210, 11nfmpt 4124 . . . . . . . . . 10  |-  F/_ x
( k  e.  _V  |->  B )
139, 5, 12nfseq 11072 . . . . . . . . 9  |-  F/_ x  seq  m ( G , 
( k  e.  _V  |->  B ) )
14 nfcv 2432 . . . . . . . . 9  |-  F/_ x n
1513, 14nffv 5548 . . . . . . . 8  |-  F/_ x
(  seq  m ( G ,  ( k  e.  _V  |->  B ) ) `
 n )
1615nfcri 2426 . . . . . . 7  |-  F/ x  z  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n )
178, 16nfan 1783 . . . . . 6  |-  F/ x
( A  =  ( m ... n )  /\  z  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  B ) ) `
 n ) )
187, 17nfrex 2611 . . . . 5  |-  F/ x E. n  e.  ( ZZ>=
`  m ) ( A  =  ( m ... n )  /\  z  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) )
1918nfex 1779 . . . 4  |-  F/ x E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  z  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  B ) ) `
 n ) )
2019nfab 2436 . . 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 3602 . 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 2429 1  |-  F/_ x prod_ k  e.  A G B
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419   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 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-3an 936  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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-seq 11063  df-prod 25402
  Copyright terms: Public domain W3C validator