Theorem nfcprod1 14640

Description: Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Hypothesis

Ref	Expression
nfcprod1.1	|- F/_ k A

Assertion

Ref	Expression
nfcprod1	|- F/_ k prod_ k e. A B

Distinct variable group:   A, k

Allowed substitution hint:   B( k)

Proof of Theorem nfcprod1

Dummy variables f m n x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prod 14636	. 2 |- prod_ k e. A B = ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ E. n e. ( ZZ>= ` m ) E. y ( y =/= 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y ) /\ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) ) )
2		nfcv 2764	. . . . 5 |- F/_ k ZZ
3		nfcprod1.1	. . . . . . 7 |- F/_ k A
4		nfcv 2764	. . . . . . 7 |- F/_ k ( ZZ>= ` m )
5	3, 4	nfss 3596	. . . . . 6 |- F/ k A C_ ( ZZ>= ` m )
6		nfv 1843	. . . . . . . . 9 |- F/ k y =/= 0
7		nfcv 2764	. . . . . . . . . . 11 |- F/_ k n
8		nfcv 2764	. . . . . . . . . . 11 |- F/_ k x.
9		nfmpt1 4747	. . . . . . . . . . 11 |- F/_ k ( k e. ZZ |-> if ( k e. A , B , 1 ) )
10	7, 8, 9	nfseq 12811	. . . . . . . . . 10 |- F/_ k seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
11		nfcv 2764	. . . . . . . . . 10 |- F/_ k ~~>
12		nfcv 2764	. . . . . . . . . 10 |- F/_ k y
13	10, 11, 12	nfbr 4699	. . . . . . . . 9 |- F/ k seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y
14	6, 13	nfan 1828	. . . . . . . 8 |- F/ k ( y =/= 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y )
15	14	nfex 2154	. . . . . . 7 |- F/ k E. y ( y =/= 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y )
16	4, 15	nfrex 3007	. . . . . 6 |- F/ k E. n e. ( ZZ>= ` m ) E. y ( y =/= 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y )
17		nfcv 2764	. . . . . . . 8 |- F/_ k m
18	17, 8, 9	nfseq 12811	. . . . . . 7 |- F/_ k seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
19		nfcv 2764	. . . . . . 7 |- F/_ k x
20	18, 11, 19	nfbr 4699	. . . . . 6 |- F/ k seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x
21	5, 16, 20	nf3an 1831	. . . . 5 |- F/ k ( A C_ ( ZZ>= ` m ) /\ E. n e. ( ZZ>= ` m ) E. y ( y =/= 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y ) /\ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x )
22	2, 21	nfrex 3007	. . . 4 |- F/ k E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ E. n e. ( ZZ>= ` m ) E. y ( y =/= 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y ) /\ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x )
23		nfcv 2764	. . . . 5 |- F/_ k NN
24		nfcv 2764	. . . . . . . 8 |- F/_ k f
25		nfcv 2764	. . . . . . . 8 |- F/_ k ( 1 ... m )
26	24, 25, 3	nff1o 6135	. . . . . . 7 |- F/ k f : ( 1 ... m ) -1-1-onto-> A
27		nfcv 2764	. . . . . . . . . 10 |- F/_ k 1
28		nfcsb1v 3549	. . . . . . . . . . 11 |- F/_ k [_ ( f ` n ) / k ]_ B
29	23, 28	nfmpt 4746	. . . . . . . . . 10 |- F/_ k ( n e. NN |-> [_ ( f ` n ) / k ]_ B )
30	27, 8, 29	nfseq 12811	. . . . . . . . 9 |- F/_ k seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) )
31	30, 17	nffv 6198	. . . . . . . 8 |- F/_ k ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m )
32	31	nfeq2 2780	. . . . . . 7 |- F/ k x = ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m )
33	26, 32	nfan 1828	. . . . . 6 |- F/ k ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) )
34	33	nfex 2154	. . . . 5 |- F/ k E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) )
35	23, 34	nfrex 3007	. . . 4 |- F/ k E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) )
36	22, 35	nfor 1834	. . 3 |- F/ k ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ E. n e. ( ZZ>= ` m ) E. y ( y =/= 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y ) /\ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) )
37	36	nfiota 5855	. 2 |- F/_ k ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ E. n e. ( ZZ>= ` m ) E. y ( y =/= 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y ) /\ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) ) ` m ) ) ) )
38	1, 37	nfcxfr 2762	1 |- F/_ k 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:  fprodcn  39832  dvmptfprod  40160  vonicc  40899