Theorem nfcprod1 11865

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 11862	. 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 2348	. . . . 5 |- F/_ k ZZ
3		nfcprod1.1	. . . . . . . 8 |- F/_ k A
4		nfcv 2348	. . . . . . . 8 |- F/_ k ( ZZ>= ` m )
5	3, 4	nfss 3186	. . . . . . 7 |- F/ k A C_ ( ZZ>= ` m )
6	3	nfcri 2342	. . . . . . . . 9 |- F/ k j e. A
7	6	nfdc 1682	. . . . . . . 8 |- F/ kDECID j e. A
8	4, 7	nfralxy 2544	. . . . . . 7 |- F/ k A. j e. ( ZZ>= ` m )DECID j e. A
9	5, 8	nfan 1588	. . . . . 6 |- F/ k ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A )
10		nfv 1551	. . . . . . . . . 10 |- F/ k y # 0
11		nfcv 2348	. . . . . . . . . . . 12 |- F/_ k n
12		nfcv 2348	. . . . . . . . . . . 12 |- F/_ k x.
13		nfmpt1 4137	. . . . . . . . . . . 12 |- F/_ k ( k e. ZZ |-> if ( k e. A , B , 1 ) )
14	11, 12, 13	nfseq 10602	. . . . . . . . . . 11 |- F/_ k seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
15		nfcv 2348	. . . . . . . . . . 11 |- F/_ k ~~>
16		nfcv 2348	. . . . . . . . . . 11 |- F/_ k y
17	14, 15, 16	nfbr 4090	. . . . . . . . . 10 |- F/ k seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y
18	10, 17	nfan 1588	. . . . . . . . 9 |- F/ k ( y # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y )
19	18	nfex 1660	. . . . . . . 8 |- F/ k E. y ( y # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y )
20	4, 19	nfrexw 2545	. . . . . . 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 2348	. . . . . . . . 9 |- F/_ k m
22	21, 12, 13	nfseq 10602	. . . . . . . 8 |- F/_ k seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
23		nfcv 2348	. . . . . . . 8 |- F/_ k x
24	22, 15, 23	nfbr 4090	. . . . . . 7 |- F/ k seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x
25	20, 24	nfan 1588	. . . . . 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 1588	. . . . 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	nfrexw 2545	. . . 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 2348	. . . . 5 |- F/_ k NN
29		nfcv 2348	. . . . . . . 8 |- F/_ k f
30		nfcv 2348	. . . . . . . 8 |- F/_ k ( 1 ... m )
31	29, 30, 3	nff1o 5520	. . . . . . 7 |- F/ k f : ( 1 ... m ) -1-1-onto-> A
32		nfcv 2348	. . . . . . . . . 10 |- F/_ k 1
33		nfv 1551	. . . . . . . . . . . 12 |- F/ k n <_ m
34		nfcsb1v 3126	. . . . . . . . . . . 12 |- F/_ k [_ ( f ` n ) / k ]_ B
35	33, 34, 32	nfif 3599	. . . . . . . . . . 11 |- F/_ k if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 )
36	28, 35	nfmpt 4136	. . . . . . . . . 10 |- F/_ k ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) )
37	32, 12, 36	nfseq 10602	. . . . . . . . 9 |- F/_ k seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) )
38	37, 21	nffv 5586	. . . . . . . 8 |- F/_ k ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
39	38	nfeq2 2360	. . . . . . 7 |- F/ k x = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
40	31, 39	nfan 1588	. . . . . 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 1660	. . . . 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	nfrexw 2545	. . . 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 1597	. . 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 5236	. 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 2345	1 |- F/_ k prod_ k e. A B

Syntax hints:   /\ wa 104   \/ wo 710  DECID wdc 836   = wceq 1373   E.wex 1515   e. wcel 2176   F/_wnfc 2335   A.wral 2484   E.wrex 2485   [_csb 3093   C_ wss 3166   ifcif 3571   class class class wbr 4044   |-> cmpt 4105   iotacio 5230   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5944   0cc0 7925   1c1 7926   x. cmul 7930   <_ cle 8108   # cap 8654   NNcn 9036   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130   seqcseq 10592   ~~> cli 11589   prod_cprod 11861

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-if 3572  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-recs 6391  df-frec 6477  df-seqfrec 10593  df-proddc 11862

This theorem is referenced by: (None)