Theorem nfcprod1 12114

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 12111	. 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 2374	. . . . 5 |- F/_ k ZZ
3		nfcprod1.1	. . . . . . . 8 |- F/_ k A
4		nfcv 2374	. . . . . . . 8 |- F/_ k ( ZZ>= ` m )
5	3, 4	nfss 3220	. . . . . . 7 |- F/ k A C_ ( ZZ>= ` m )
6	3	nfcri 2368	. . . . . . . . 9 |- F/ k j e. A
7	6	nfdc 1707	. . . . . . . 8 |- F/ kDECID j e. A
8	4, 7	nfralxy 2570	. . . . . . 7 |- F/ k A. j e. ( ZZ>= ` m )DECID j e. A
9	5, 8	nfan 1613	. . . . . 6 |- F/ k ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A )
10		nfv 1576	. . . . . . . . . 10 |- F/ k y # 0
11		nfcv 2374	. . . . . . . . . . . 12 |- F/_ k n
12		nfcv 2374	. . . . . . . . . . . 12 |- F/_ k x.
13		nfmpt1 4182	. . . . . . . . . . . 12 |- F/_ k ( k e. ZZ |-> if ( k e. A , B , 1 ) )
14	11, 12, 13	nfseq 10718	. . . . . . . . . . 11 |- F/_ k seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
15		nfcv 2374	. . . . . . . . . . 11 |- F/_ k ~~>
16		nfcv 2374	. . . . . . . . . . 11 |- F/_ k y
17	14, 15, 16	nfbr 4135	. . . . . . . . . 10 |- F/ k seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y
18	10, 17	nfan 1613	. . . . . . . . 9 |- F/ k ( y # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y )
19	18	nfex 1685	. . . . . . . 8 |- F/ k E. y ( y # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y )
20	4, 19	nfrexw 2571	. . . . . . 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 2374	. . . . . . . . 9 |- F/_ k m
22	21, 12, 13	nfseq 10718	. . . . . . . 8 |- F/_ k seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) )
23		nfcv 2374	. . . . . . . 8 |- F/_ k x
24	22, 15, 23	nfbr 4135	. . . . . . 7 |- F/ k seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x
25	20, 24	nfan 1613	. . . . . 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 1613	. . . . 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 2571	. . . 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 2374	. . . . 5 |- F/_ k NN
29		nfcv 2374	. . . . . . . 8 |- F/_ k f
30		nfcv 2374	. . . . . . . 8 |- F/_ k ( 1 ... m )
31	29, 30, 3	nff1o 5581	. . . . . . 7 |- F/ k f : ( 1 ... m ) -1-1-onto-> A
32		nfcv 2374	. . . . . . . . . 10 |- F/_ k 1
33		nfv 1576	. . . . . . . . . . . 12 |- F/ k n <_ m
34		nfcsb1v 3160	. . . . . . . . . . . 12 |- F/_ k [_ ( f ` n ) / k ]_ B
35	33, 34, 32	nfif 3634	. . . . . . . . . . 11 |- F/_ k if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 )
36	28, 35	nfmpt 4181	. . . . . . . . . 10 |- F/_ k ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) )
37	32, 12, 36	nfseq 10718	. . . . . . . . 9 |- F/_ k seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) )
38	37, 21	nffv 5649	. . . . . . . 8 |- F/_ k ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
39	38	nfeq2 2386	. . . . . . 7 |- F/ k x = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m )
40	31, 39	nfan 1613	. . . . . 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 1685	. . . . 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 2571	. . . 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 1622	. . 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 5290	. 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 2371	1 |- F/_ k prod_ k e. A B

Syntax hints:   /\ wa 104   \/ wo 715  DECID wdc 841   = wceq 1397   E.wex 1540   e. wcel 2202   F/_wnfc 2361   A.wral 2510   E.wrex 2511   [_csb 3127   C_ wss 3200   ifcif 3605   class class class wbr 4088   |-> cmpt 4150   iotacio 5284   -1-1-onto->wf1o 5325   ` cfv 5326  (class class class)co 6017   0cc0 8031   1c1 8032   x. cmul 8036   <_ cle 8214   # cap 8760   NNcn 9142   ZZcz 9478   ZZ>=cuz 9754   ...cfz 10242   seqcseq 10708   ~~> cli 11838   prod_cprod 12110

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-recs 6470  df-frec 6556  df-seqfrec 10709  df-proddc 12111

This theorem is referenced by: (None)