Theorem nfsum 11382

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 11379	. 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 2331	. . . . 5 |- F/_ x ZZ
3		nfsum.1	. . . . . . 7 |- F/_ x A
4		nfcv 2331	. . . . . . 7 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 3162	. . . . . 6 |- F/ x A C_ ( ZZ>= ` m )
6	3	nfcri 2325	. . . . . . . 8 |- F/ x j e. A
7	6	nfdc 1669	. . . . . . 7 |- F/ xDECID j e. A
8	4, 7	nfralxy 2527	. . . . . 6 |- F/ x A. j e. ( ZZ>= ` m )DECID j e. A
9		nfcv 2331	. . . . . . . 8 |- F/_ x m
10		nfcv 2331	. . . . . . . 8 |- F/_ x +
11	3	nfcri 2325	. . . . . . . . . 10 |- F/ x n e. A
12		nfcv 2331	. . . . . . . . . . 11 |- F/_ x n
13		nfsum.2	. . . . . . . . . . 11 |- F/_ x B
14	12, 13	nfcsb 3108	. . . . . . . . . 10 |- F/_ x [_ n / k ]_ B
15		nfcv 2331	. . . . . . . . . 10 |- F/_ x 0
16	11, 14, 15	nfif 3576	. . . . . . . . 9 |- F/_ x if ( n e. A , [_ n / k ]_ B , 0 )
17	2, 16	nfmpt 4109	. . . . . . . 8 |- F/_ x ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
18	9, 10, 17	nfseq 10472	. . . . . . 7 |- F/_ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
19		nfcv 2331	. . . . . . 7 |- F/_ x ~~>
20		nfcv 2331	. . . . . . 7 |- F/_ x z
21	18, 19, 20	nfbr 4063	. . . . . 6 |- F/ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> z
22	5, 8, 21	nf3an 1576	. . . . 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	nfrexxy 2528	. . . 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 2331	. . . . 5 |- F/_ x NN
25		nfcv 2331	. . . . . . . 8 |- F/_ x f
26		nfcv 2331	. . . . . . . 8 |- F/_ x ( 1 ... m )
27	25, 26, 3	nff1o 5473	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
28		nfcv 2331	. . . . . . . . . 10 |- F/_ x 1
29		nfv 1538	. . . . . . . . . . . 12 |- F/ x n <_ m
30		nfcv 2331	. . . . . . . . . . . . 13 |- F/_ x ( f ` n )
31	30, 13	nfcsb 3108	. . . . . . . . . . . 12 |- F/_ x [_ ( f ` n ) / k ]_ B
32	29, 31, 15	nfif 3576	. . . . . . . . . . 11 |- F/_ x if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
33	24, 32	nfmpt 4109	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
34	28, 10, 33	nfseq 10472	. . . . . . . . 9 |- F/_ x seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) )
35	34, 9	nffv 5539	. . . . . . . 8 |- F/_ x ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
36	35	nfeq2 2343	. . . . . . 7 |- F/ x z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
37	27, 36	nfan 1575	. . . . . 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 1647	. . . . 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	nfrexxy 2528	. . . 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 1584	. . 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 5196	. 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 2328	1 |- F/_ x sum_ k e. A B

Syntax hints:   /\ wa 104   \/ wo 709  DECID wdc 835   /\ w3a 979   = wceq 1363   E.wex 1502   e. wcel 2159   F/_wnfc 2318   A.wral 2467   E.wrex 2468   [_csb 3071   C_ wss 3143   ifcif 3548   class class class wbr 4017   |-> cmpt 4078   iotacio 5190   -1-1-onto->wf1o 5229   ` cfv 5230  (class class class)co 5890   0cc0 7828   1c1 7829   + caddc 7831   <_ cle 8010   NNcn 8936   ZZcz 9270   ZZ>=cuz 9545   ...cfz 10025   seqcseq 10462   ~~> cli 11303   sum_csu 11378

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-un 3147  df-in 3149  df-ss 3156  df-if 3549  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-mpt 4080  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-ov 5893  df-oprab 5894  df-mpo 5895  df-recs 6323  df-frec 6409  df-seqfrec 10463  df-sumdc 11379

This theorem is referenced by:  fsum2dlemstep  11459  fisumcom2  11463  fsumiun  11502  fsumcncntop  14439