Theorem nfcprod 11356

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 11352	. 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 2282	. . . . 5 |- F/_ x ZZ
3		nfcprod.1	. . . . . . . 8 |- F/_ x A
4		nfcv 2282	. . . . . . . 8 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 3095	. . . . . . 7 |- F/ x A C_ ( ZZ>= ` m )
6	3	nfcri 2276	. . . . . . . . 9 |- F/ x j e. A
7	6	nfdc 1638	. . . . . . . 8 |- F/ xDECID j e. A
8	4, 7	nfralxy 2474	. . . . . . 7 |- F/ x A. j e. ( ZZ>= ` m )DECID j e. A
9	5, 8	nfan 1545	. . . . . 6 |- F/ x ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A )
10		nfv 1509	. . . . . . . . . 10 |- F/ x z # 0
11		nfcv 2282	. . . . . . . . . . . 12 |- F/_ x n
12		nfcv 2282	. . . . . . . . . . . 12 |- F/_ x x.
13	3	nfcri 2276	. . . . . . . . . . . . . 14 |- F/ x k e. A
14		nfcprod.2	. . . . . . . . . . . . . 14 |- F/_ x B
15		nfcv 2282	. . . . . . . . . . . . . 14 |- F/_ x 1
16	13, 14, 15	nfif 3505	. . . . . . . . . . . . 13 |- F/_ x if ( k e. A , B , 1 )
17	2, 16	nfmpt 4028	. . . . . . . . . . . 12 |- F/_ x ( k e. ZZ |-> if ( k e. A , B , 1 ) )
18	11, 12, 17	nfseq 10259	. . . . . . . . . . 11 |- F/_ x seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
19		nfcv 2282	. . . . . . . . . . 11 |- F/_ x ~~>
20		nfcv 2282	. . . . . . . . . . 11 |- F/_ x z
21	18, 19, 20	nfbr 3982	. . . . . . . . . 10 |- F/ x seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z
22	10, 21	nfan 1545	. . . . . . . . 9 |- F/ x ( z # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z )
23	22	nfex 1617	. . . . . . . 8 |- F/ x E. z ( z # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z )
24	4, 23	nfrexxy 2475	. . . . . . 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 2282	. . . . . . . . 9 |- F/_ x m
26	25, 12, 17	nfseq 10259	. . . . . . . 8 |- F/_ x seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
27		nfcv 2282	. . . . . . . 8 |- F/_ x y
28	26, 19, 27	nfbr 3982	. . . . . . 7 |- F/ x seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y
29	24, 28	nfan 1545	. . . . . 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 1545	. . . . 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	nfrexxy 2475	. . . 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 2282	. . . . 5 |- F/_ x NN
33		nfcv 2282	. . . . . . . 8 |- F/_ x f
34		nfcv 2282	. . . . . . . 8 |- F/_ x ( 1 ... m )
35	33, 34, 3	nff1o 5373	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
36		nfv 1509	. . . . . . . . . . . 12 |- F/ x n <_ m
37		nfcv 2282	. . . . . . . . . . . . 13 |- F/_ x ( f ` n )
38	37, 14	nfcsb 3042	. . . . . . . . . . . 12 |- F/_ x [_ ( f ` n ) / k ]_ B
39	36, 38, 15	nfif 3505	. . . . . . . . . . 11 |- F/_ x if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 )
40	32, 39	nfmpt 4028	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) )
41	15, 12, 40	nfseq 10259	. . . . . . . . 9 |- F/_ x seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) )
42	41, 25	nffv 5439	. . . . . . . 8 |- F/_ x ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
43	42	nfeq2 2294	. . . . . . 7 |- F/ x y = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
44	35, 43	nfan 1545	. . . . . 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 1617	. . . . 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	nfrexxy 2475	. . . 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 1554	. . 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 5100	. 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 2279	1 |- F/_ x prod_ k e. A B

Syntax hints:   /\ wa 103   \/ wo 698  DECID wdc 820   = wceq 1332   E.wex 1469   e. wcel 1481   F/_wnfc 2269   A.wral 2417   E.wrex 2418   [_csb 3007   C_ wss 3076   ifcif 3479   class class class wbr 3937   |-> cmpt 3997   iotacio 5094   -1-1-onto->wf1o 5130   ` cfv 5131  (class class class)co 5782   0cc0 7644   1c1 7645   x. cmul 7649   <_ cle 7825   # cap 8367   NNcn 8744   ZZcz 9078   ZZ>=cuz 9350   ...cfz 9821   seqcseq 10249   ~~> cli 11079   prod_cprod 11351

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-if 3480  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-recs 6210  df-frec 6296  df-seqfrec 10250  df-proddc 11352

This theorem is referenced by: (None)