Theorem nfcprod 12066

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 12062	. 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 2372	. . . . 5 |- F/_ x ZZ
3		nfcprod.1	. . . . . . . 8 |- F/_ x A
4		nfcv 2372	. . . . . . . 8 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 3217	. . . . . . 7 |- F/ x A C_ ( ZZ>= ` m )
6	3	nfcri 2366	. . . . . . . . 9 |- F/ x j e. A
7	6	nfdc 1705	. . . . . . . 8 |- F/ xDECID j e. A
8	4, 7	nfralxy 2568	. . . . . . 7 |- F/ x A. j e. ( ZZ>= ` m )DECID j e. A
9	5, 8	nfan 1611	. . . . . 6 |- F/ x ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A )
10		nfv 1574	. . . . . . . . . 10 |- F/ x z # 0
11		nfcv 2372	. . . . . . . . . . . 12 |- F/_ x n
12		nfcv 2372	. . . . . . . . . . . 12 |- F/_ x x.
13	3	nfcri 2366	. . . . . . . . . . . . . 14 |- F/ x k e. A
14		nfcprod.2	. . . . . . . . . . . . . 14 |- F/_ x B
15		nfcv 2372	. . . . . . . . . . . . . 14 |- F/_ x 1
16	13, 14, 15	nfif 3631	. . . . . . . . . . . . 13 |- F/_ x if ( k e. A , B , 1 )
17	2, 16	nfmpt 4176	. . . . . . . . . . . 12 |- F/_ x ( k e. ZZ |-> if ( k e. A , B , 1 ) )
18	11, 12, 17	nfseq 10679	. . . . . . . . . . 11 |- F/_ x seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
19		nfcv 2372	. . . . . . . . . . 11 |- F/_ x ~~>
20		nfcv 2372	. . . . . . . . . . 11 |- F/_ x z
21	18, 19, 20	nfbr 4130	. . . . . . . . . 10 |- F/ x seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z
22	10, 21	nfan 1611	. . . . . . . . 9 |- F/ x ( z # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z )
23	22	nfex 1683	. . . . . . . 8 |- F/ x E. z ( z # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z )
24	4, 23	nfrexw 2569	. . . . . . 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 2372	. . . . . . . . 9 |- F/_ x m
26	25, 12, 17	nfseq 10679	. . . . . . . 8 |- F/_ x seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
27		nfcv 2372	. . . . . . . 8 |- F/_ x y
28	26, 19, 27	nfbr 4130	. . . . . . 7 |- F/ x seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y
29	24, 28	nfan 1611	. . . . . 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 1611	. . . . 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 2569	. . . 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 2372	. . . . 5 |- F/_ x NN
33		nfcv 2372	. . . . . . . 8 |- F/_ x f
34		nfcv 2372	. . . . . . . 8 |- F/_ x ( 1 ... m )
35	33, 34, 3	nff1o 5570	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
36		nfv 1574	. . . . . . . . . . . 12 |- F/ x n <_ m
37		nfcv 2372	. . . . . . . . . . . . 13 |- F/_ x ( f ` n )
38	37, 14	nfcsb 3162	. . . . . . . . . . . 12 |- F/_ x [_ ( f ` n ) / k ]_ B
39	36, 38, 15	nfif 3631	. . . . . . . . . . 11 |- F/_ x if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 )
40	32, 39	nfmpt 4176	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) )
41	15, 12, 40	nfseq 10679	. . . . . . . . 9 |- F/_ x seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) )
42	41, 25	nffv 5637	. . . . . . . 8 |- F/_ x ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
43	42	nfeq2 2384	. . . . . . 7 |- F/ x y = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
44	35, 43	nfan 1611	. . . . . 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 1683	. . . . 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 2569	. . . 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 1620	. . 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 5282	. 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 2369	1 |- F/_ x prod_ k e. A B

Syntax hints:   /\ wa 104   \/ wo 713  DECID wdc 839   = wceq 1395   E.wex 1538   e. wcel 2200   F/_wnfc 2359   A.wral 2508   E.wrex 2509   [_csb 3124   C_ wss 3197   ifcif 3602   class class class wbr 4083   |-> cmpt 4145   iotacio 5276   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6001   0cc0 7999   1c1 8000   x. cmul 8004   <_ cle 8182   # cap 8728   NNcn 9110   ZZcz 9446   ZZ>=cuz 9722   ...cfz 10204   seqcseq 10669   ~~> cli 11789   prod_cprod 12061

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-recs 6451  df-frec 6537  df-seqfrec 10670  df-proddc 12062

This theorem is referenced by:  fprod2dlemstep  12133  fprodcom2fi  12137