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Theorem nfcprod 28896
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 28891 . 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 2629 . . . . 5  |-  F/_ x ZZ
3 nfcprod.1 . . . . . . 7  |-  F/_ x A
4 nfcv 2629 . . . . . . 7  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 3497 . . . . . 6  |-  F/ x  A  C_  ( ZZ>= `  m
)
6 nfv 1683 . . . . . . . . 9  |-  F/ x  z  =/=  0
7 nfcv 2629 . . . . . . . . . . 11  |-  F/_ x n
8 nfcv 2629 . . . . . . . . . . 11  |-  F/_ x  x.
93nfcri 2622 . . . . . . . . . . . . 13  |-  F/ x  k  e.  A
10 nfcprod.2 . . . . . . . . . . . . 13  |-  F/_ x B
11 nfcv 2629 . . . . . . . . . . . . 13  |-  F/_ x
1
129, 10, 11nfif 3968 . . . . . . . . . . . 12  |-  F/_ x if ( k  e.  A ,  B ,  1 )
132, 12nfmpt 4535 . . . . . . . . . . 11  |-  F/_ x
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
147, 8, 13nfseq 12086 . . . . . . . . . 10  |-  F/_ x  seq n (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )
15 nfcv 2629 . . . . . . . . . 10  |-  F/_ x  ~~>
16 nfcv 2629 . . . . . . . . . 10  |-  F/_ x
z
1714, 15, 16nfbr 4491 . . . . . . . . 9  |-  F/ x  seq n (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z
186, 17nfan 1875 . . . . . . . 8  |-  F/ x
( z  =/=  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )
1918nfex 1895 . . . . . . 7  |-  F/ x E. z ( z  =/=  0  /\  seq n
(  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )
204, 19nfrex 2927 . . . . . 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 2629 . . . . . . . 8  |-  F/_ x m
2221, 8, 13nfseq 12086 . . . . . . 7  |-  F/_ x  seq m (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )
23 nfcv 2629 . . . . . . 7  |-  F/_ x
y
2422, 15, 23nfbr 4491 . . . . . 6  |-  F/ x  seq m (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y
255, 20, 24nf3an 1877 . . . . 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 2927 . . . 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 2629 . . . . 5  |-  F/_ x NN
28 nfcv 2629 . . . . . . . 8  |-  F/_ x
f
29 nfcv 2629 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
3028, 29, 3nff1o 5814 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
31 nfcv 2629 . . . . . . . . . . . 12  |-  F/_ x
( f `  n
)
3231, 10nfcsb 3453 . . . . . . . . . . 11  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
3327, 32nfmpt 4535 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
3411, 8, 33nfseq 12086 . . . . . . . . 9  |-  F/_ x  seq 1 (  x.  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
3534, 21nffv 5873 . . . . . . . 8  |-  F/_ x
(  seq 1 (  x.  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m )
3635nfeq2 2646 . . . . . . 7  |-  F/ x  y  =  (  seq 1 (  x.  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
3730, 36nfan 1875 . . . . . 6  |-  F/ x
( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq 1 (  x.  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3837nfex 1895 . . . . 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 2927 . . . 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 1882 . . 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 5555 . 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 2627 1  |-  F/_ x prod_ k  e.  A  B
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   F/_wnfc 2615    =/= wne 2662   E.wrex 2815   [_csb 3435    C_ wss 3476   ifcif 3939   class class class wbr 4447    |-> cmpt 4505   iotacio 5549   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6285   0cc0 9493   1c1 9494    x. cmul 9498   NNcn 10537   ZZcz 10865   ZZ>=cuz 11083   ...cfz 11673    seqcseq 12076    ~~> cli 13273   prod_cprod 28890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-recs 7043  df-rdg 7077  df-seq 12077  df-prod 28891
This theorem is referenced by:  fprod2dlem  28963  fprodcom2  28967
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