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Theorem nfcprod 13908
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 13903 . 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 2569 . . . . 5  |-  F/_ x ZZ
3 nfcprod.1 . . . . . . 7  |-  F/_ x A
4 nfcv 2569 . . . . . . 7  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 3400 . . . . . 6  |-  F/ x  A  C_  ( ZZ>= `  m
)
6 nfv 1755 . . . . . . . . 9  |-  F/ x  z  =/=  0
7 nfcv 2569 . . . . . . . . . . 11  |-  F/_ x n
8 nfcv 2569 . . . . . . . . . . 11  |-  F/_ x  x.
93nfcri 2563 . . . . . . . . . . . . 13  |-  F/ x  k  e.  A
10 nfcprod.2 . . . . . . . . . . . . 13  |-  F/_ x B
11 nfcv 2569 . . . . . . . . . . . . 13  |-  F/_ x
1
129, 10, 11nfif 3883 . . . . . . . . . . . 12  |-  F/_ x if ( k  e.  A ,  B ,  1 )
132, 12nfmpt 4455 . . . . . . . . . . 11  |-  F/_ x
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
147, 8, 13nfseq 12173 . . . . . . . . . 10  |-  F/_ x  seq n (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )
15 nfcv 2569 . . . . . . . . . 10  |-  F/_ x  ~~>
16 nfcv 2569 . . . . . . . . . 10  |-  F/_ x
z
1714, 15, 16nfbr 4411 . . . . . . . . 9  |-  F/ x  seq n (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z
186, 17nfan 1988 . . . . . . . 8  |-  F/ x
( z  =/=  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )
1918nfex 2008 . . . . . . 7  |-  F/ x E. z ( z  =/=  0  /\  seq n
(  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )
204, 19nfrex 2827 . . . . . 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 2569 . . . . . . . 8  |-  F/_ x m
2221, 8, 13nfseq 12173 . . . . . . 7  |-  F/_ x  seq m (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )
23 nfcv 2569 . . . . . . 7  |-  F/_ x
y
2422, 15, 23nfbr 4411 . . . . . 6  |-  F/ x  seq m (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y
255, 20, 24nf3an 1990 . . . . 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 2827 . . . 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 2569 . . . . 5  |-  F/_ x NN
28 nfcv 2569 . . . . . . . 8  |-  F/_ x
f
29 nfcv 2569 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
3028, 29, 3nff1o 5772 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
31 nfcv 2569 . . . . . . . . . . . 12  |-  F/_ x
( f `  n
)
3231, 10nfcsb 3356 . . . . . . . . . . 11  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
3327, 32nfmpt 4455 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
3411, 8, 33nfseq 12173 . . . . . . . . 9  |-  F/_ x  seq 1 (  x.  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
3534, 21nffv 5832 . . . . . . . 8  |-  F/_ x
(  seq 1 (  x.  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m )
3635nfeq2 2584 . . . . . . 7  |-  F/ x  y  =  (  seq 1 (  x.  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
3730, 36nfan 1988 . . . . . 6  |-  F/ x
( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq 1 (  x.  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3837nfex 2008 . . . . 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 2827 . . . 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 1995 . . 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 5512 . 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 2567 1  |-  F/_ x prod_ k  e.  A  B
Colors of variables: wff setvar class
Syntax hints:    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872   F/_wnfc 2556    =/= wne 2599   E.wrex 2715   [_csb 3338    C_ wss 3379   ifcif 3854   class class class wbr 4366    |-> cmpt 4425   iotacio 5506   -1-1-onto->wf1o 5543   ` cfv 5544  (class class class)co 6249   0cc0 9490   1c1 9491    x. cmul 9495   NNcn 10560   ZZcz 10888   ZZ>=cuz 11110   ...cfz 11735    seqcseq 12163    ~~> cli 13491   prod_cprod 13902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-seq 12164  df-prod 13903
This theorem is referenced by:  fprod2dlem  13977  fprodcom2  13981  fprodcncf  37662
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