Theorem nfcprod 12196

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 12192	. 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 2375	. . . . 5 |- F/_ x ZZ
3		nfcprod.1	. . . . . . . 8 |- F/_ x A
4		nfcv 2375	. . . . . . . 8 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 3221	. . . . . . 7 |- F/ x A C_ ( ZZ>= ` m )
6	3	nfcri 2369	. . . . . . . . 9 |- F/ x j e. A
7	6	nfdc 1707	. . . . . . . 8 |- F/ xDECID j e. A
8	4, 7	nfralxy 2571	. . . . . . 7 |- F/ x A. j e. ( ZZ>= ` m )DECID j e. A
9	5, 8	nfan 1614	. . . . . 6 |- F/ x ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A )
10		nfv 1577	. . . . . . . . . 10 |- F/ x z # 0
11		nfcv 2375	. . . . . . . . . . . 12 |- F/_ x n
12		nfcv 2375	. . . . . . . . . . . 12 |- F/_ x x.
13	3	nfcri 2369	. . . . . . . . . . . . . 14 |- F/ x k e. A
14		nfcprod.2	. . . . . . . . . . . . . 14 |- F/_ x B
15		nfcv 2375	. . . . . . . . . . . . . 14 |- F/_ x 1
16	13, 14, 15	nfif 3638	. . . . . . . . . . . . 13 |- F/_ x if ( k e. A , B , 1 )
17	2, 16	nfmpt 4186	. . . . . . . . . . . 12 |- F/_ x ( k e. ZZ |-> if ( k e. A , B , 1 ) )
18	11, 12, 17	nfseq 10782	. . . . . . . . . . 11 |- F/_ x seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
19		nfcv 2375	. . . . . . . . . . 11 |- F/_ x ~~>
20		nfcv 2375	. . . . . . . . . . 11 |- F/_ x z
21	18, 19, 20	nfbr 4140	. . . . . . . . . 10 |- F/ x seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z
22	10, 21	nfan 1614	. . . . . . . . 9 |- F/ x ( z # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z )
23	22	nfex 1686	. . . . . . . 8 |- F/ x E. z ( z # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z )
24	4, 23	nfrexw 2572	. . . . . . 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 2375	. . . . . . . . 9 |- F/_ x m
26	25, 12, 17	nfseq 10782	. . . . . . . 8 |- F/_ x seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
27		nfcv 2375	. . . . . . . 8 |- F/_ x y
28	26, 19, 27	nfbr 4140	. . . . . . 7 |- F/ x seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y
29	24, 28	nfan 1614	. . . . . 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 1614	. . . . 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 2572	. . . 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 2375	. . . . 5 |- F/_ x NN
33		nfcv 2375	. . . . . . . 8 |- F/_ x f
34		nfcv 2375	. . . . . . . 8 |- F/_ x ( 1 ... m )
35	33, 34, 3	nff1o 5590	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
36		nfv 1577	. . . . . . . . . . . 12 |- F/ x n <_ m
37		nfcv 2375	. . . . . . . . . . . . 13 |- F/_ x ( f ` n )
38	37, 14	nfcsb 3166	. . . . . . . . . . . 12 |- F/_ x [_ ( f ` n ) / k ]_ B
39	36, 38, 15	nfif 3638	. . . . . . . . . . 11 |- F/_ x if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 )
40	32, 39	nfmpt 4186	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) )
41	15, 12, 40	nfseq 10782	. . . . . . . . 9 |- F/_ x seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) )
42	41, 25	nffv 5658	. . . . . . . 8 |- F/_ x ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
43	42	nfeq2 2387	. . . . . . 7 |- F/ x y = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
44	35, 43	nfan 1614	. . . . . 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 1686	. . . . 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 2572	. . . 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 1623	. . 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 5297	. 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 2372	1 |- F/_ x prod_ k e. A B

Syntax hints:   /\ wa 104   \/ wo 716  DECID wdc 842   = wceq 1398   E.wex 1541   e. wcel 2202   F/_wnfc 2362   A.wral 2511   E.wrex 2512   [_csb 3128   C_ wss 3201   ifcif 3607   class class class wbr 4093   |-> cmpt 4155   iotacio 5291   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   0cc0 8092   1c1 8093   x. cmul 8097   <_ cle 8274   # cap 8820   NNcn 9202   ZZcz 9540   ZZ>=cuz 9816   ...cfz 10305   seqcseq 10772   ~~> cli 11918   prod_cprod 12191

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-recs 6514  df-frec 6600  df-seqfrec 10773  df-proddc 12192

This theorem is referenced by:  fprod2dlemstep  12263  fprodcom2fi  12267