Theorem nfcprod 11698

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.

Step	Hyp	Ref	Expression
1		df-proddc 11694	. 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 2336	. . . . 5 |- F/_ x ZZ
3		nfcprod.1	. . . . . . . 8 |- F/_ x A
4		nfcv 2336	. . . . . . . 8 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 3172	. . . . . . 7 |- F/ x A C_ ( ZZ>= ` m )
6	3	nfcri 2330	. . . . . . . . 9 |- F/ x j e. A
7	6	nfdc 1670	. . . . . . . 8 |- F/ xDECID j e. A
8	4, 7	nfralxy 2532	. . . . . . 7 |- F/ x A. j e. ( ZZ>= ` m )DECID j e. A
9	5, 8	nfan 1576	. . . . . 6 |- F/ x ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A )
10		nfv 1539	. . . . . . . . . 10 |- F/ x z # 0
11		nfcv 2336	. . . . . . . . . . . 12 |- F/_ x n
12		nfcv 2336	. . . . . . . . . . . 12 |- F/_ x x.
13	3	nfcri 2330	. . . . . . . . . . . . . 14 |- F/ x k e. A
14		nfcprod.2	. . . . . . . . . . . . . 14 |- F/_ x B
15		nfcv 2336	. . . . . . . . . . . . . 14 |- F/_ x 1
16	13, 14, 15	nfif 3585	. . . . . . . . . . . . 13 |- F/_ x if ( k e. A , B , 1 )
17	2, 16	nfmpt 4121	. . . . . . . . . . . 12 |- F/_ x ( k e. ZZ |-> if ( k e. A , B , 1 ) )
18	11, 12, 17	nfseq 10528	. . . . . . . . . . 11 |- F/_ x seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
19		nfcv 2336	. . . . . . . . . . 11 |- F/_ x ~~>
20		nfcv 2336	. . . . . . . . . . 11 |- F/_ x z
21	18, 19, 20	nfbr 4075	. . . . . . . . . 10 |- F/ x seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z
22	10, 21	nfan 1576	. . . . . . . . 9 |- F/ x ( z # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z )
23	22	nfex 1648	. . . . . . . 8 |- F/ x E. z ( z # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z )
24	4, 23	nfrexw 2533	. . . . . . 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 2336	. . . . . . . . 9 |- F/_ x m
26	25, 12, 17	nfseq 10528	. . . . . . . 8 |- F/_ x seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
27		nfcv 2336	. . . . . . . 8 |- F/_ x y
28	26, 19, 27	nfbr 4075	. . . . . . 7 |- F/ x seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y
29	24, 28	nfan 1576	. . . . . 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 )
30	9, 29	nfan 1576	. . . . 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 ) )
31	2, 30	nfrexw 2533	. . . 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 2336	. . . . 5 |- F/_ x NN
33		nfcv 2336	. . . . . . . 8 |- F/_ x f
34		nfcv 2336	. . . . . . . 8 |- F/_ x ( 1 ... m )
35	33, 34, 3	nff1o 5498	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
36		nfv 1539	. . . . . . . . . . . 12 |- F/ x n <_ m
37		nfcv 2336	. . . . . . . . . . . . 13 |- F/_ x ( f ` n )
38	37, 14	nfcsb 3118	. . . . . . . . . . . 12 |- F/_ x [_ ( f ` n ) / k ]_ B
39	36, 38, 15	nfif 3585	. . . . . . . . . . 11 |- F/_ x if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 )
40	32, 39	nfmpt 4121	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) )
41	15, 12, 40	nfseq 10528	. . . . . . . . 9 |- F/_ x seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) )
42	41, 25	nffv 5564	. . . . . . . 8 |- F/_ x ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
43	42	nfeq2 2348	. . . . . . 7 |- F/ x y = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
44	35, 43	nfan 1576	. . . . . 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 ) )
45	44	nfex 1648	. . . . 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 ) )
46	32, 45	nfrexw 2533	. . . 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 ) )
47	31, 46	nfor 1585	. . 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 ) ) )
48	47	nfiotaw 5219	. 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 ) ) ) )
49	1, 48	nfcxfr 2333	1 |- F/_ x prod_ k e. A B

Syntax hints:   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   E.wex 1503   e. wcel 2164   F/_wnfc 2323   A.wral 2472   E.wrex 2473   [_csb 3080   C_ wss 3153   ifcif 3557   class class class wbr 4029   |-> cmpt 4090   iotacio 5213   -1-1-onto->wf1o 5253   ` cfv 5254  (class class class)co 5918   0cc0 7872   1c1 7873   x. cmul 7877   <_ cle 8055   # cap 8600   NNcn 8982   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074   seqcseq 10518   ~~> cli 11421   prod_cprod 11693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-recs 6358  df-frec 6444  df-seqfrec 10519  df-proddc 11694

This theorem is referenced by:  fprod2dlemstep  11765  fprodcom2fi  11769