Theorem nfsum 10809

Description: Bound-variable hypothesis builder for sum: if x is (effectively) not free in A and B, it is not free in sum_ k e. A B. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Hypotheses

Ref	Expression
nfsum.1	|- F/_ x A
nfsum.2	|- F/_ x B

Assertion

Ref	Expression
nfsum	|- F/_ x sum_ k e. A B

Proof of Theorem nfsum

Dummy variables f j m n z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-isum 10806	. 2 |- sum_ k e. A B = ( iota z ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> z ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) , CC ) ` m ) ) ) )
2		nfcv 2229	. . . . 5 |- F/_ x ZZ
3		nfsum.1	. . . . . . 7 |- F/_ x A
4		nfcv 2229	. . . . . . 7 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 3021	. . . . . 6 |- F/ x A C_ ( ZZ>= ` m )
6	3	nfcri 2223	. . . . . . . 8 |- F/ x j e. A
7	6	nfdc 1595	. . . . . . 7 |- F/ xDECID j e. A
8	4, 7	nfralxy 2415	. . . . . 6 |- F/ x A. j e. ( ZZ>= ` m )DECID j e. A
9		nfcv 2229	. . . . . . . 8 |- F/_ x m
10		nfcv 2229	. . . . . . . 8 |- F/_ x +
11	3	nfcri 2223	. . . . . . . . . 10 |- F/ x n e. A
12		nfcv 2229	. . . . . . . . . . 11 |- F/_ x n
13		nfsum.2	. . . . . . . . . . 11 |- F/_ x B
14	12, 13	nfcsb 2968	. . . . . . . . . 10 |- F/_ x [_ n / k ]_ B
15		nfcv 2229	. . . . . . . . . 10 |- F/_ x 0
16	11, 14, 15	nfif 3425	. . . . . . . . 9 |- F/_ x if ( n e. A , [_ n / k ]_ B , 0 )
17	2, 16	nfmpt 3938	. . . . . . . 8 |- F/_ x ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
18		nfcv 2229	. . . . . . . 8 |- F/_ x CC
19	9, 10, 17, 18	nfiseq 9931	. . . . . . 7 |- F/_ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC )
20		nfcv 2229	. . . . . . 7 |- F/_ x ~~>
21		nfcv 2229	. . . . . . 7 |- F/_ x z
22	19, 20, 21	nfbr 3897	. . . . . 6 |- F/ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> z
23	5, 8, 22	nf3an 1504	. . . . 5 |- F/ x ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> z )
24	2, 23	nfrexxy 2416	. . . 4 |- F/ x E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> z )
25		nfcv 2229	. . . . 5 |- F/_ x NN
26		nfcv 2229	. . . . . . . 8 |- F/_ x f
27		nfcv 2229	. . . . . . . 8 |- F/_ x ( 1 ... m )
28	26, 27, 3	nff1o 5266	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
29		nfcv 2229	. . . . . . . . . 10 |- F/_ x 1
30		nfv 1467	. . . . . . . . . . . 12 |- F/ x n <_ m
31		nfcv 2229	. . . . . . . . . . . . 13 |- F/_ x ( f ` n )
32	31, 13	nfcsb 2968	. . . . . . . . . . . 12 |- F/_ x [_ ( f ` n ) / k ]_ B
33	30, 32, 15	nfif 3425	. . . . . . . . . . 11 |- F/_ x if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
34	25, 33	nfmpt 3938	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
35	29, 10, 34, 18	nfiseq 9931	. . . . . . . . 9 |- F/_ x seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) , CC )
36	35, 9	nffv 5330	. . . . . . . 8 |- F/_ x ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) , CC ) ` m )
37	36	nfeq2 2241	. . . . . . 7 |- F/ x z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) , CC ) ` m )
38	28, 37	nfan 1503	. . . . . 6 |- F/ x ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) , CC ) ` m ) )
39	38	nfex 1574	. . . . 5 |- F/ x E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) , CC ) ` m ) )
40	25, 39	nfrexxy 2416	. . . 4 |- F/ x E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) , CC ) ` m ) )
41	24, 40	nfor 1512	. . 3 |- F/ x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> z ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) , CC ) ` m ) ) )
42	41	nfiotaxy 4999	. 2 |- F/_ x ( iota z ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> z ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) , CC ) ` m ) ) ) )
43	1, 42	nfcxfr 2226	1 |- F/_ x sum_ k e. A B

Syntax hints:   /\ wa 103   \/ wo 665  DECID wdc 781   /\ w3a 925   = wceq 1290   E.wex 1427   e. wcel 1439   F/_wnfc 2216   A.wral 2360   E.wrex 2361   [_csb 2936   C_ wss 3002   ifcif 3399   class class class wbr 3853   |-> cmpt 3907   iotacio 4993   -1-1-onto->wf1o 5029   ` cfv 5030  (class class class)co 5668   CCcc 7411   0cc0 7413   1c1 7414   + caddc 7416   <_ cle 7586   NNcn 8485   ZZcz 8813   ZZ>=cuz 9082   ...cfz 9487   seqcseq4 9914   ~~> cli 10729   sum_csu 10805

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-un 3006  df-in 3008  df-ss 3015  df-if 3400  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-recs 6086  df-frec 6172  df-iseq 9916  df-isum 10806

This theorem is referenced by:  fsum2dlemstep  10891  fisumcom2  10895  fsumiun  10934