Theorem nfcprod1 11517

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

Step	Hyp	Ref	Expression
1		df-proddc 11514	. 2 |- prod_ k e. A B = ( iota x ( E. m e. ZZ ( ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A ) /\ ( 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 |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m ) ) ) )
2		nfcv 2312	. . . . 5 |- F/_ k ZZ
3		nfcprod1.1	. . . . . . . 8 |- F/_ k A
4		nfcv 2312	. . . . . . . 8 |- F/_ k ( ZZ>= ` m )
5	3, 4	nfss 3140	. . . . . . 7 |- F/ k A C_ ( ZZ>= ` m )
6	3	nfcri 2306	. . . . . . . . 9 |- F/ k j e. A
7	6	nfdc 1652	. . . . . . . 8 |- F/ kDECID j e. A
8	4, 7	nfralxy 2508	. . . . . . 7 |- F/ k A. j e. ( ZZ>= ` m )DECID j e. A
9	5, 8	nfan 1558	. . . . . 6 |- F/ k ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A )
10		nfv 1521	. . . . . . . . . 10 |- F/ k y # 0
11		nfcv 2312	. . . . . . . . . . . 12 |- F/_ k n
12		nfcv 2312	. . . . . . . . . . . 12 |- F/_ k x.
13		nfmpt1 4082	. . . . . . . . . . . 12 |- F/_ k ( k e. ZZ |-> if ( k e. A , B , 1 ) )
14	11, 12, 13	nfseq 10411	. . . . . . . . . . 11 |- F/_ k seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
15		nfcv 2312	. . . . . . . . . . 11 |- F/_ k ~~>
16		nfcv 2312	. . . . . . . . . . 11 |- F/_ k y
17	14, 15, 16	nfbr 4035	. . . . . . . . . 10 |- F/ k seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y
18	10, 17	nfan 1558	. . . . . . . . 9 |- F/ k ( y # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y )
19	18	nfex 1630	. . . . . . . 8 |- F/ k E. y ( y # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y )
20	4, 19	nfrexxy 2509	. . . . . . 7 |- F/ k E. n e. ( ZZ>= ` m ) E. y ( y # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y )
21		nfcv 2312	. . . . . . . . 9 |- F/_ k m
22	21, 12, 13	nfseq 10411	. . . . . . . 8 |- F/_ k seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
23		nfcv 2312	. . . . . . . 8 |- F/_ k x
24	22, 15, 23	nfbr 4035	. . . . . . 7 |- F/ k seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x
25	20, 24	nfan 1558	. . . . . 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 ) /\ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x )
26	9, 25	nfan 1558	. . . . 5 |- F/ k ( ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A ) /\ ( 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 ) )
27	2, 26	nfrexxy 2509	. . . 4 |- F/ k E. m e. ZZ ( ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A ) /\ ( 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 ) )
28		nfcv 2312	. . . . 5 |- F/_ k NN
29		nfcv 2312	. . . . . . . 8 |- F/_ k f
30		nfcv 2312	. . . . . . . 8 |- F/_ k ( 1 ... m )
31	29, 30, 3	nff1o 5440	. . . . . . 7 |- F/ k f : ( 1 ... m ) -1-1-onto-> A
32		nfcv 2312	. . . . . . . . . 10 |- F/_ k 1
33		nfv 1521	. . . . . . . . . . . 12 |- F/ k n <_ m
34		nfcsb1v 3082	. . . . . . . . . . . 12 |- F/_ k [_ ( f ` n ) / k ]_ B
35	33, 34, 32	nfif 3554	. . . . . . . . . . 11 |- F/_ k if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 )
36	28, 35	nfmpt 4081	. . . . . . . . . 10 |- F/_ k ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) )
37	32, 12, 36	nfseq 10411	. . . . . . . . 9 |- F/_ k seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) )
38	37, 21	nffv 5506	. . . . . . . 8 |- F/_ k ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
39	38	nfeq2 2324	. . . . . . 7 |- F/ k x = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
40	31, 39	nfan 1558	. . . . . 6 |- F/ k ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m ) )
41	40	nfex 1630	. . . . 5 |- F/ k E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m ) )
42	28, 41	nfrexxy 2509	. . . 4 |- F/ k E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m ) )
43	27, 42	nfor 1567	. . 3 |- F/ k ( E. m e. ZZ ( ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A ) /\ ( 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 |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m ) ) )
44	43	nfiotaw 5164	. 2 |- F/_ k ( iota x ( E. m e. ZZ ( ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A ) /\ ( 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 |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m ) ) ) )
45	1, 44	nfcxfr 2309	1 |- F/_ k prod_ k e. A B

Syntax hints:   /\ wa 103   \/ wo 703  DECID wdc 829   = wceq 1348   E.wex 1485   e. wcel 2141   F/_wnfc 2299   A.wral 2448   E.wrex 2449   [_csb 3049   C_ wss 3121   ifcif 3526   class class class wbr 3989   |-> cmpt 4050   iotacio 5158   -1-1-onto->wf1o 5197   ` cfv 5198  (class class class)co 5853   0cc0 7774   1c1 7775   x. cmul 7779   <_ cle 7955   # cap 8500   NNcn 8878   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   seqcseq 10401   ~~> cli 11241   prod_cprod 11513

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-recs 6284  df-frec 6370  df-seqfrec 10402  df-proddc 11514

This theorem is referenced by: (None)