Theorem nfcprod 14641

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 m n y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prod 14636	. 2 |- prod_ k e. A B = ( iota y ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ 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 |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) ) )
2		nfcv 2764	. . . . 5 |- F/_ x ZZ
3		nfcprod.1	. . . . . . 7 |- F/_ x A
4		nfcv 2764	. . . . . . 7 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 3596	. . . . . 6 |- F/ x A C_ ( ZZ>= ` m )
6		nfv 1843	. . . . . . . . 9 |- F/ x z =/= 0
7		nfcv 2764	. . . . . . . . . . 11 |- F/_ x n
8		nfcv 2764	. . . . . . . . . . 11 |- F/_ x x.
9	3	nfcri 2758	. . . . . . . . . . . . 13 |- F/ x k e. A
10		nfcprod.2	. . . . . . . . . . . . 13 |- F/_ x B
11		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ x 1
12	9, 10, 11	nfif 4115	. . . . . . . . . . . 12 |- F/_ x if ( k e. A , B , 1 )
13	2, 12	nfmpt 4746	. . . . . . . . . . 11 |- F/_ x ( k e. ZZ |-> if ( k e. A , B , 1 ) )
14	7, 8, 13	nfseq 12811	. . . . . . . . . 10 |- F/_ x seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
15		nfcv 2764	. . . . . . . . . 10 |- F/_ x ~~>
16		nfcv 2764	. . . . . . . . . 10 |- F/_ x z
17	14, 15, 16	nfbr 4699	. . . . . . . . 9 |- F/ x seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z
18	6, 17	nfan 1828	. . . . . . . 8 |- F/ x ( z =/= 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z )
19	18	nfex 2154	. . . . . . 7 |- F/ x E. z ( z =/= 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> z )
20	4, 19	nfrex 3007	. . . . . 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 )
21		nfcv 2764	. . . . . . . 8 |- F/_ x m
22	21, 8, 13	nfseq 12811	. . . . . . 7 |- F/_ x seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
23		nfcv 2764	. . . . . . 7 |- F/_ x y
24	22, 15, 23	nfbr 4699	. . . . . 6 |- F/ x seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y
25	5, 20, 24	nf3an 1831	. . . . 5 |- F/ x ( A C_ ( ZZ>= ` m ) /\ 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 )
26	2, 25	nfrex 3007	. . . 4 |- F/ x E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ 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 )
27		nfcv 2764	. . . . 5 |- F/_ x NN
28		nfcv 2764	. . . . . . . 8 |- F/_ x f
29		nfcv 2764	. . . . . . . 8 |- F/_ x ( 1 ... m )
30	28, 29, 3	nff1o 6135	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
31		nfcv 2764	. . . . . . . . . . . 12 |- F/_ x ( f ` n )
32	31, 10	nfcsb 3551	. . . . . . . . . . 11 |- F/_ x [_ ( f ` n ) / k ]_ B
33	27, 32	nfmpt 4746	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> [_ ( f ` n ) / k ]_ B )
34	11, 8, 33	nfseq 12811	. . . . . . . . 9 |- F/_ x seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) )
35	34, 21	nffv 6198	. . . . . . . 8 |- F/_ x ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m )
36	35	nfeq2 2780	. . . . . . 7 |- F/ x y = ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m )
37	30, 36	nfan 1828	. . . . . 6 |- F/ x ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) )
38	37	nfex 2154	. . . . 5 |- F/ x E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) )
39	27, 38	nfrex 3007	. . . 4 |- F/ x E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) )
40	26, 39	nfor 1834	. . 3 |- F/ x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ 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 |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) )
41	40	nfiota 5855	. 2 |- F/_ x ( iota y ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ 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 |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) ) )
42	1, 41	nfcxfr 2762	1 |- F/_ x prod_ k e. A B

Syntax hints:   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   E.wrex 2913   [_csb 3533   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   iotacio 5849   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ~~> cli 14215   prod_cprod 14635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802  df-prod 14636

This theorem is referenced by:  fprod2dlem  14710  fprodcom2  14714  fprodcom2OLD  14715  fprodcn  39832  fprodcncf  40114