Theorem nfsum 11503

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-sumdc 11500	. 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 ) ) ) ~~> 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 ) ) ) ` m ) ) ) )
2		nfcv 2336	. . . . 5 |- F/_ x ZZ
3		nfsum.1	. . . . . . 7 |- F/_ x A
4		nfcv 2336	. . . . . . 7 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 3173	. . . . . 6 |- F/ x A C_ ( ZZ>= ` m )
6	3	nfcri 2330	. . . . . . . 8 |- F/ x j e. A
7	6	nfdc 1670	. . . . . . 7 |- F/ xDECID j e. A
8	4, 7	nfralxy 2532	. . . . . 6 |- F/ x A. j e. ( ZZ>= ` m )DECID j e. A
9		nfcv 2336	. . . . . . . 8 |- F/_ x m
10		nfcv 2336	. . . . . . . 8 |- F/_ x +
11	3	nfcri 2330	. . . . . . . . . 10 |- F/ x n e. A
12		nfcv 2336	. . . . . . . . . . 11 |- F/_ x n
13		nfsum.2	. . . . . . . . . . 11 |- F/_ x B
14	12, 13	nfcsb 3119	. . . . . . . . . 10 |- F/_ x [_ n / k ]_ B
15		nfcv 2336	. . . . . . . . . 10 |- F/_ x 0
16	11, 14, 15	nfif 3586	. . . . . . . . 9 |- F/_ x if ( n e. A , [_ n / k ]_ B , 0 )
17	2, 16	nfmpt 4122	. . . . . . . 8 |- F/_ x ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
18	9, 10, 17	nfseq 10531	. . . . . . 7 |- F/_ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
19		nfcv 2336	. . . . . . 7 |- F/_ x ~~>
20		nfcv 2336	. . . . . . 7 |- F/_ x z
21	18, 19, 20	nfbr 4076	. . . . . 6 |- F/ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> z
22	5, 8, 21	nf3an 1577	. . . . 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 ) ) ) ~~> z )
23	2, 22	nfrexw 2533	. . . 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 ) ) ) ~~> z )
24		nfcv 2336	. . . . 5 |- F/_ x NN
25		nfcv 2336	. . . . . . . 8 |- F/_ x f
26		nfcv 2336	. . . . . . . 8 |- F/_ x ( 1 ... m )
27	25, 26, 3	nff1o 5499	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
28		nfcv 2336	. . . . . . . . . 10 |- F/_ x 1
29		nfv 1539	. . . . . . . . . . . 12 |- F/ x n <_ m
30		nfcv 2336	. . . . . . . . . . . . 13 |- F/_ x ( f ` n )
31	30, 13	nfcsb 3119	. . . . . . . . . . . 12 |- F/_ x [_ ( f ` n ) / k ]_ B
32	29, 31, 15	nfif 3586	. . . . . . . . . . 11 |- F/_ x if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
33	24, 32	nfmpt 4122	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
34	28, 10, 33	nfseq 10531	. . . . . . . . 9 |- F/_ x seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) )
35	34, 9	nffv 5565	. . . . . . . 8 |- F/_ x ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
36	35	nfeq2 2348	. . . . . . 7 |- F/ x z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
37	27, 36	nfan 1576	. . . . . 6 |- F/ x ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
38	37	nfex 1648	. . . . 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 ) ) ) ` m ) )
39	24, 38	nfrexw 2533	. . . 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 ) ) ) ` m ) )
40	23, 39	nfor 1585	. . 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 ) ) ) ~~> 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 ) ) ) ` m ) ) )
41	40	nfiotaw 5220	. 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 ) ) ) ~~> 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 ) ) ) ` m ) ) ) )
42	1, 41	nfcxfr 2333	1 |- F/_ x sum_ k e. A B

Syntax hints:   /\ wa 104   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   F/_wnfc 2323   A.wral 2472   E.wrex 2473   [_csb 3081   C_ wss 3154   ifcif 3558   class class class wbr 4030   |-> cmpt 4091   iotacio 5214   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5919   0cc0 7874   1c1 7875   + caddc 7877   <_ cle 8057   NNcn 8984   ZZcz 9320   ZZ>=cuz 9595   ...cfz 10077   seqcseq 10521   ~~> cli 11424   sum_csu 11499

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-if 3559  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-recs 6360  df-frec 6446  df-seqfrec 10522  df-sumdc 11500

This theorem is referenced by:  fsum2dlemstep  11580  fisumcom2  11584  fsumiun  11623  fsumcncntop  14746  dvmptfsum  14904