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Theorem nfcprod 13800
Description: Bound-variable hypothesis builder for product: if  x is (effectively) not free in  A and  B, it is not free in  prod_ k  e.  A B. (Contributed by Scott Fenton, 1-Dec-2017.)
Hypotheses
Ref Expression
nfcprod.1  |-  F/_ x A
nfcprod.2  |-  F/_ x B
Assertion
Ref Expression
nfcprod  |-  F/_ x prod_ k  e.  A  B
Distinct variable group:    x, k
Allowed substitution hints:    A( x, k)    B( x, k)

Proof of Theorem nfcprod
Dummy variables  f  m  n  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-prod 13795 . 2  |-  prod_ k  e.  A  B  =  ( iota y ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  E. n  e.  ( ZZ>= `  m ) E. z ( z  =/=  0  /\  seq n
(  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )  /\  seq m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq 1 (  x.  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
2 nfcv 2616 . . . . 5  |-  F/_ x ZZ
3 nfcprod.1 . . . . . . 7  |-  F/_ x A
4 nfcv 2616 . . . . . . 7  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 3482 . . . . . 6  |-  F/ x  A  C_  ( ZZ>= `  m
)
6 nfv 1712 . . . . . . . . 9  |-  F/ x  z  =/=  0
7 nfcv 2616 . . . . . . . . . . 11  |-  F/_ x n
8 nfcv 2616 . . . . . . . . . . 11  |-  F/_ x  x.
93nfcri 2609 . . . . . . . . . . . . 13  |-  F/ x  k  e.  A
10 nfcprod.2 . . . . . . . . . . . . 13  |-  F/_ x B
11 nfcv 2616 . . . . . . . . . . . . 13  |-  F/_ x
1
129, 10, 11nfif 3958 . . . . . . . . . . . 12  |-  F/_ x if ( k  e.  A ,  B ,  1 )
132, 12nfmpt 4527 . . . . . . . . . . 11  |-  F/_ x
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
147, 8, 13nfseq 12099 . . . . . . . . . 10  |-  F/_ x  seq n (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )
15 nfcv 2616 . . . . . . . . . 10  |-  F/_ x  ~~>
16 nfcv 2616 . . . . . . . . . 10  |-  F/_ x
z
1714, 15, 16nfbr 4483 . . . . . . . . 9  |-  F/ x  seq n (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z
186, 17nfan 1933 . . . . . . . 8  |-  F/ x
( z  =/=  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )
1918nfex 1953 . . . . . . 7  |-  F/ x E. z ( z  =/=  0  /\  seq n
(  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )
204, 19nfrex 2917 . . . . . 6  |-  F/ x E. n  e.  ( ZZ>=
`  m ) E. z ( z  =/=  0  /\  seq n
(  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )
21 nfcv 2616 . . . . . . . 8  |-  F/_ x m
2221, 8, 13nfseq 12099 . . . . . . 7  |-  F/_ x  seq m (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )
23 nfcv 2616 . . . . . . 7  |-  F/_ x
y
2422, 15, 23nfbr 4483 . . . . . 6  |-  F/ x  seq m (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y
255, 20, 24nf3an 1935 . . . . 5  |-  F/ x
( A  C_  ( ZZ>=
`  m )  /\  E. n  e.  ( ZZ>= `  m ) E. z
( z  =/=  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )  /\  seq m (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )
262, 25nfrex 2917 . . . 4  |-  F/ x E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  E. n  e.  ( ZZ>= `  m ) E. z
( z  =/=  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )  /\  seq m (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )
27 nfcv 2616 . . . . 5  |-  F/_ x NN
28 nfcv 2616 . . . . . . . 8  |-  F/_ x
f
29 nfcv 2616 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
3028, 29, 3nff1o 5796 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
31 nfcv 2616 . . . . . . . . . . . 12  |-  F/_ x
( f `  n
)
3231, 10nfcsb 3438 . . . . . . . . . . 11  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
3327, 32nfmpt 4527 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
3411, 8, 33nfseq 12099 . . . . . . . . 9  |-  F/_ x  seq 1 (  x.  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
3534, 21nffv 5855 . . . . . . . 8  |-  F/_ x
(  seq 1 (  x.  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m )
3635nfeq2 2633 . . . . . . 7  |-  F/ x  y  =  (  seq 1 (  x.  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
3730, 36nfan 1933 . . . . . 6  |-  F/ x
( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq 1 (  x.  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3837nfex 1953 . . . . 5  |-  F/ x E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  y  =  (  seq 1 (  x.  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3927, 38nfrex 2917 . . . 4  |-  F/ x E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  y  =  (  seq 1 (  x.  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
4026, 39nfor 1940 . . 3  |-  F/ x
( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  E. n  e.  (
ZZ>= `  m ) E. z ( z  =/=  0  /\  seq n
(  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )  /\  seq m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq 1 (  x.  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) )
4140nfiota 5538 . 2  |-  F/_ x
( iota y ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  E. n  e.  ( ZZ>= `  m ) E. z ( z  =/=  0  /\  seq n
(  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )  /\  seq m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq 1 (  x.  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
421, 41nfcxfr 2614 1  |-  F/_ x prod_ k  e.  A  B
Colors of variables: wff setvar class
Syntax hints:    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823   F/_wnfc 2602    =/= wne 2649   E.wrex 2805   [_csb 3420    C_ wss 3461   ifcif 3929   class class class wbr 4439    |-> cmpt 4497   iotacio 5532   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    x. cmul 9486   NNcn 10531   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675    seqcseq 12089    ~~> cli 13389   prod_cprod 13794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-recs 7034  df-rdg 7068  df-seq 12090  df-prod 13795
This theorem is referenced by:  fprod2dlem  13866  fprodcom2  13870  fprodcncf  31943
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