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Theorem nfcprod 11331
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  j  m  n  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-proddc 11327 . 2  |-  prod_ k  e.  A  B  =  ( iota y ( E. m  e.  ZZ  (
( A  C_  ( ZZ>=
`  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A )  /\  ( 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  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 ) ) ) `  m
) ) ) )
2 nfcv 2281 . . . . 5  |-  F/_ x ZZ
3 nfcprod.1 . . . . . . . 8  |-  F/_ x A
4 nfcv 2281 . . . . . . . 8  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 3090 . . . . . . 7  |-  F/ x  A  C_  ( ZZ>= `  m
)
63nfcri 2275 . . . . . . . . 9  |-  F/ x  j  e.  A
76nfdc 1637 . . . . . . . 8  |-  F/ xDECID  j  e.  A
84, 7nfralxy 2471 . . . . . . 7  |-  F/ x A. j  e.  ( ZZ>=
`  m )DECID  j  e.  A
95, 8nfan 1544 . . . . . 6  |-  F/ x
( A  C_  ( ZZ>=
`  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A )
10 nfv 1508 . . . . . . . . . 10  |-  F/ x  z #  0
11 nfcv 2281 . . . . . . . . . . . 12  |-  F/_ x n
12 nfcv 2281 . . . . . . . . . . . 12  |-  F/_ x  x.
133nfcri 2275 . . . . . . . . . . . . . 14  |-  F/ x  k  e.  A
14 nfcprod.2 . . . . . . . . . . . . . 14  |-  F/_ x B
15 nfcv 2281 . . . . . . . . . . . . . 14  |-  F/_ x
1
1613, 14, 15nfif 3500 . . . . . . . . . . . . 13  |-  F/_ x if ( k  e.  A ,  B ,  1 )
172, 16nfmpt 4020 . . . . . . . . . . . 12  |-  F/_ x
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
1811, 12, 17nfseq 10235 . . . . . . . . . . 11  |-  F/_ x  seq n (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )
19 nfcv 2281 . . . . . . . . . . 11  |-  F/_ x  ~~>
20 nfcv 2281 . . . . . . . . . . 11  |-  F/_ x
z
2118, 19, 20nfbr 3974 . . . . . . . . . 10  |-  F/ x  seq n (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z
2210, 21nfan 1544 . . . . . . . . 9  |-  F/ x
( z #  0  /\ 
seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )
2322nfex 1616 . . . . . . . 8  |-  F/ x E. z ( z #  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )
244, 23nfrexxy 2472 . . . . . . 7  |-  F/ x E. n  e.  ( ZZ>=
`  m ) E. z ( z #  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  z )
25 nfcv 2281 . . . . . . . . 9  |-  F/_ x m
2625, 12, 17nfseq 10235 . . . . . . . 8  |-  F/_ x  seq m (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )
27 nfcv 2281 . . . . . . . 8  |-  F/_ x
y
2826, 19, 27nfbr 3974 . . . . . . 7  |-  F/ x  seq m (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y
2924, 28nfan 1544 . . . . . 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 )  /\  seq m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )
309, 29nfan 1544 . . . . 5  |-  F/ x
( ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  (
ZZ>= `  m )DECID  j  e.  A )  /\  ( 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 ) )
312, 30nfrexxy 2472 . . . 4  |-  F/ x E. m  e.  ZZ  ( ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  (
ZZ>= `  m )DECID  j  e.  A )  /\  ( 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 ) )
32 nfcv 2281 . . . . 5  |-  F/_ x NN
33 nfcv 2281 . . . . . . . 8  |-  F/_ x
f
34 nfcv 2281 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
3533, 34, 3nff1o 5365 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
36 nfv 1508 . . . . . . . . . . . 12  |-  F/ x  n  <_  m
37 nfcv 2281 . . . . . . . . . . . . 13  |-  F/_ x
( f `  n
)
3837, 14nfcsb 3037 . . . . . . . . . . . 12  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
3936, 38, 15nfif 3500 . . . . . . . . . . 11  |-  F/_ x if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 )
4032, 39nfmpt 4020 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 ) )
4115, 12, 40nfseq 10235 . . . . . . . . 9  |-  F/_ x  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 ) ) )
4241, 25nffv 5431 . . . . . . . 8  |-  F/_ x
(  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  1 ) ) ) `  m
)
4342nfeq2 2293 . . . . . . 7  |-  F/ x  y  =  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 ) ) ) `  m
)
4435, 43nfan 1544 . . . . . 6  |-  F/ x
( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  1 ) ) ) `  m
) )
4544nfex 1616 . . . . 5  |-  F/ x E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  y  =  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  1 ) ) ) `  m
) )
4632, 45nfrexxy 2472 . . . 4  |-  F/ x E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  y  =  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  1 ) ) ) `  m
) )
4731, 46nfor 1553 . . 3  |-  F/ x
( E. m  e.  ZZ  ( ( A 
C_  ( ZZ>= `  m
)  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A )  /\  ( 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  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 ) ) ) `  m
) ) )
4847nfiotaw 5092 . 2  |-  F/_ x
( iota y ( E. m  e.  ZZ  (
( A  C_  ( ZZ>=
`  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A )  /\  ( 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  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 ) ) ) `  m
) ) ) )
491, 48nfcxfr 2278 1  |-  F/_ x prod_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 697  DECID wdc 819    = wceq 1331   E.wex 1468    e. wcel 1480   F/_wnfc 2268   A.wral 2416   E.wrex 2417   [_csb 3003    C_ wss 3071   ifcif 3474   class class class wbr 3929    |-> cmpt 3989   iotacio 5086   -1-1-onto->wf1o 5122   ` cfv 5123  (class class class)co 5774   0cc0 7627   1c1 7628    x. cmul 7632    <_ cle 7808   # cap 8350   NNcn 8727   ZZcz 9061   ZZ>=cuz 9333   ...cfz 9797    seqcseq 10225    ~~> cli 11054   prod_cprod 11326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-seqfrec 10226  df-proddc 11327
This theorem is referenced by: (None)
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