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Theorem nfcprod1 11330
Description: Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypothesis
Ref Expression
nfcprod1.1  |-  F/_ k A
Assertion
Ref Expression
nfcprod1  |-  F/_ k prod_ k  e.  A  B
Distinct variable group:    A, k
Allowed substitution hint:    B( k)

Proof of Theorem nfcprod1
Dummy variables  f  j  m  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-proddc 11327 . 2  |-  prod_ k  e.  A  B  =  ( iota x ( E. m  e.  ZZ  (
( A  C_  ( ZZ>=
`  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A )  /\  ( E. n  e.  ( ZZ>= `  m ) E. y ( y #  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )  /\  seq m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  x ) )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 ) ) ) `  m
) ) ) )
2 nfcv 2281 . . . . 5  |-  F/_ k ZZ
3 nfcprod1.1 . . . . . . . 8  |-  F/_ k A
4 nfcv 2281 . . . . . . . 8  |-  F/_ k
( ZZ>= `  m )
53, 4nfss 3090 . . . . . . 7  |-  F/ k  A  C_  ( ZZ>= `  m )
63nfcri 2275 . . . . . . . . 9  |-  F/ k  j  e.  A
76nfdc 1637 . . . . . . . 8  |-  F/ kDECID  j  e.  A
84, 7nfralxy 2471 . . . . . . 7  |-  F/ k A. j  e.  (
ZZ>= `  m )DECID  j  e.  A
95, 8nfan 1544 . . . . . 6  |-  F/ k ( A  C_  ( ZZ>=
`  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A )
10 nfv 1508 . . . . . . . . . 10  |-  F/ k  y #  0
11 nfcv 2281 . . . . . . . . . . . 12  |-  F/_ k
n
12 nfcv 2281 . . . . . . . . . . . 12  |-  F/_ k  x.
13 nfmpt1 4021 . . . . . . . . . . . 12  |-  F/_ k
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
1411, 12, 13nfseq 10235 . . . . . . . . . . 11  |-  F/_ k  seq n (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )
15 nfcv 2281 . . . . . . . . . . 11  |-  F/_ k  ~~>
16 nfcv 2281 . . . . . . . . . . 11  |-  F/_ k
y
1714, 15, 16nfbr 3974 . . . . . . . . . 10  |-  F/ k  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y
1810, 17nfan 1544 . . . . . . . . 9  |-  F/ k ( y #  0  /\ 
seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )
1918nfex 1616 . . . . . . . 8  |-  F/ k E. y ( y #  0  /\  seq n
(  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )
204, 19nfrexxy 2472 . . . . . . 7  |-  F/ k E. n  e.  (
ZZ>= `  m ) E. y ( y #  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )
21 nfcv 2281 . . . . . . . . 9  |-  F/_ k
m
2221, 12, 13nfseq 10235 . . . . . . . 8  |-  F/_ k  seq m (  x.  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )
23 nfcv 2281 . . . . . . . 8  |-  F/_ k
x
2422, 15, 23nfbr 3974 . . . . . . 7  |-  F/ k  seq m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  x
2520, 24nfan 1544 . . . . . 6  |-  F/ k ( E. n  e.  ( ZZ>= `  m ) E. y ( y #  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )  /\  seq m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  x )
269, 25nfan 1544 . . . . 5  |-  F/ k ( ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  (
ZZ>= `  m )DECID  j  e.  A )  /\  ( E. n  e.  ( ZZ>=
`  m ) E. y ( y #  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )  /\  seq m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  x ) )
272, 26nfrexxy 2472 . . . 4  |-  F/ k E. m  e.  ZZ  ( ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  (
ZZ>= `  m )DECID  j  e.  A )  /\  ( E. n  e.  ( ZZ>=
`  m ) E. y ( y #  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )  /\  seq m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  x ) )
28 nfcv 2281 . . . . 5  |-  F/_ k NN
29 nfcv 2281 . . . . . . . 8  |-  F/_ k
f
30 nfcv 2281 . . . . . . . 8  |-  F/_ k
( 1 ... m
)
3129, 30, 3nff1o 5365 . . . . . . 7  |-  F/ k  f : ( 1 ... m ) -1-1-onto-> A
32 nfcv 2281 . . . . . . . . . 10  |-  F/_ k
1
33 nfv 1508 . . . . . . . . . . . 12  |-  F/ k  n  <_  m
34 nfcsb1v 3035 . . . . . . . . . . . 12  |-  F/_ k [_ ( f `  n
)  /  k ]_ B
3533, 34, 32nfif 3500 . . . . . . . . . . 11  |-  F/_ k if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 )
3628, 35nfmpt 4020 . . . . . . . . . 10  |-  F/_ k
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 ) )
3732, 12, 36nfseq 10235 . . . . . . . . 9  |-  F/_ k  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 ) ) )
3837, 21nffv 5431 . . . . . . . 8  |-  F/_ k
(  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  1 ) ) ) `  m
)
3938nfeq2 2293 . . . . . . 7  |-  F/ k  x  =  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 ) ) ) `  m
)
4031, 39nfan 1544 . . . . . 6  |-  F/ k ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  1 ) ) ) `  m
) )
4140nfex 1616 . . . . 5  |-  F/ k E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  1 ) ) ) `  m
) )
4228, 41nfrexxy 2472 . . . 4  |-  F/ k E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq 1 (  x.  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  1 ) ) ) `  m
) )
4327, 42nfor 1553 . . 3  |-  F/ k ( E. m  e.  ZZ  ( ( A 
C_  ( ZZ>= `  m
)  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A )  /\  ( E. n  e.  ( ZZ>=
`  m ) E. y ( y #  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )  /\  seq m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  x ) )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 ) ) ) `  m
) ) )
4443nfiotaw 5092 . 2  |-  F/_ k
( iota x ( E. m  e.  ZZ  (
( A  C_  ( ZZ>=
`  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A )  /\  ( E. n  e.  ( ZZ>= `  m ) E. y ( y #  0  /\  seq n (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  y )  /\  seq m (  x.  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) ) )  ~~>  x ) )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  x.  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  1 ) ) ) `  m
) ) ) )
451, 44nfcxfr 2278 1  |-  F/_ k 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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