Theorem nfcprod1 11495

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 11492	. 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 2308	. . . . 5 |- F/_ k ZZ
3		nfcprod1.1	. . . . . . . 8 |- F/_ k A
4		nfcv 2308	. . . . . . . 8 |- F/_ k ( ZZ>= ` m )
5	3, 4	nfss 3135	. . . . . . 7 |- F/ k A C_ ( ZZ>= ` m )
6	3	nfcri 2302	. . . . . . . . 9 |- F/ k j e. A
7	6	nfdc 1647	. . . . . . . 8 |- F/ kDECID j e. A
8	4, 7	nfralxy 2504	. . . . . . 7 |- F/ k A. j e. ( ZZ>= ` m )DECID j e. A
9	5, 8	nfan 1553	. . . . . 6 |- F/ k ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A )
10		nfv 1516	. . . . . . . . . 10 |- F/ k y # 0
11		nfcv 2308	. . . . . . . . . . . 12 |- F/_ k n
12		nfcv 2308	. . . . . . . . . . . 12 |- F/_ k x.
13		nfmpt1 4075	. . . . . . . . . . . 12 |- F/_ k ( k e. ZZ |-> if ( k e. A , B , 1 ) )
14	11, 12, 13	nfseq 10390	. . . . . . . . . . 11 |- F/_ k seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
15		nfcv 2308	. . . . . . . . . . 11 |- F/_ k ~~>
16		nfcv 2308	. . . . . . . . . . 11 |- F/_ k y
17	14, 15, 16	nfbr 4028	. . . . . . . . . 10 |- F/ k seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y
18	10, 17	nfan 1553	. . . . . . . . 9 |- F/ k ( y # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y )
19	18	nfex 1625	. . . . . . . 8 |- F/ k E. y ( y # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y )
20	4, 19	nfrexxy 2505	. . . . . . 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 2308	. . . . . . . . 9 |- F/_ k m
22	21, 12, 13	nfseq 10390	. . . . . . . 8 |- F/_ k seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
23		nfcv 2308	. . . . . . . 8 |- F/_ k x
24	22, 15, 23	nfbr 4028	. . . . . . 7 |- F/ k seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x
25	20, 24	nfan 1553	. . . . . 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 1553	. . . . 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 2505	. . . 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 2308	. . . . 5 |- F/_ k NN
29		nfcv 2308	. . . . . . . 8 |- F/_ k f
30		nfcv 2308	. . . . . . . 8 |- F/_ k ( 1 ... m )
31	29, 30, 3	nff1o 5430	. . . . . . 7 |- F/ k f : ( 1 ... m ) -1-1-onto-> A
32		nfcv 2308	. . . . . . . . . 10 |- F/_ k 1
33		nfv 1516	. . . . . . . . . . . 12 |- F/ k n <_ m
34		nfcsb1v 3078	. . . . . . . . . . . 12 |- F/_ k [_ ( f ` n ) / k ]_ B
35	33, 34, 32	nfif 3548	. . . . . . . . . . 11 |- F/_ k if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 )
36	28, 35	nfmpt 4074	. . . . . . . . . 10 |- F/_ k ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) )
37	32, 12, 36	nfseq 10390	. . . . . . . . 9 |- F/_ k seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) )
38	37, 21	nffv 5496	. . . . . . . 8 |- F/_ k ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
39	38	nfeq2 2320	. . . . . . 7 |- F/ k x = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
40	31, 39	nfan 1553	. . . . . 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 1625	. . . . 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 2505	. . . 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 1562	. . 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 5157	. 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 2305	1 |- F/_ k prod_ k e. A B

Syntax hints:   /\ wa 103   \/ wo 698  DECID wdc 824   = wceq 1343   E.wex 1480   e. wcel 2136   F/_wnfc 2295   A.wral 2444   E.wrex 2445   [_csb 3045   C_ wss 3116   ifcif 3520   class class class wbr 3982   |-> cmpt 4043   iotacio 5151   -1-1-onto->wf1o 5187   ` cfv 5188  (class class class)co 5842   0cc0 7753   1c1 7754   x. cmul 7758   <_ cle 7934   # cap 8479   NNcn 8857   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944   seqcseq 10380   ~~> cli 11219   prod_cprod 11491

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-recs 6273  df-frec 6359  df-seqfrec 10381  df-proddc 11492

This theorem is referenced by: (None)