Theorem nfsum 10332

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 10329	. 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 2220	. . . . 5 |- F/_ x ZZ
3		nfsum.1	. . . . . . 7 |- F/_ x A
4		nfcv 2220	. . . . . . 7 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 2993	. . . . . 6 |- F/ x A C_ ( ZZ>= ` m )
6	3	nfcri 2214	. . . . . . . 8 |- F/ x j e. A
7	6	nfdc 1590	. . . . . . 7 |- F/ xDECID j e. A
8	4, 7	nfralxy 2403	. . . . . 6 |- F/ x A. j e. ( ZZ>= ` m )DECID j e. A
9		nfcv 2220	. . . . . . . 8 |- F/_ x m
10		nfcv 2220	. . . . . . . 8 |- F/_ x +
11	3	nfcri 2214	. . . . . . . . . 10 |- F/ x n e. A
12		nfcv 2220	. . . . . . . . . . 11 |- F/_ x n
13		nfsum.2	. . . . . . . . . . 11 |- F/_ x B
14	12, 13	nfcsb 2941	. . . . . . . . . 10 |- F/_ x [_ n / k ]_ B
15		nfcv 2220	. . . . . . . . . 10 |- F/_ x 0
16	11, 14, 15	nfif 3385	. . . . . . . . 9 |- F/_ x if ( n e. A , [_ n / k ]_ B , 0 )
17	2, 16	nfmpt 3878	. . . . . . . 8 |- F/_ x ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
18		nfcv 2220	. . . . . . . 8 |- F/_ x CC
19	9, 10, 17, 18	nfiseq 9528	. . . . . . 7 |- F/_ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC )
20		nfcv 2220	. . . . . . 7 |- F/_ x ~~>
21		nfcv 2220	. . . . . . 7 |- F/_ x z
22	19, 20, 21	nfbr 3837	. . . . . 6 |- F/ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> z
23	5, 8, 22	nf3an 1499	. . . . 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 2404	. . . 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 2220	. . . . 5 |- F/_ x NN
26		nfcv 2220	. . . . . . . 8 |- F/_ x f
27		nfcv 2220	. . . . . . . 8 |- F/_ x ( 1 ... m )
28	26, 27, 3	nff1o 5155	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
29		nfcv 2220	. . . . . . . . . 10 |- F/_ x 1
30		nfv 1462	. . . . . . . . . . . 12 |- F/ x n <_ m
31		nfcv 2220	. . . . . . . . . . . . 13 |- F/_ x ( f ` n )
32	31, 13	nfcsb 2941	. . . . . . . . . . . 12 |- F/_ x [_ ( f ` n ) / k ]_ B
33	30, 32, 15	nfif 3385	. . . . . . . . . . 11 |- F/_ x if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
34	25, 33	nfmpt 3878	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
35	29, 10, 34, 18	nfiseq 9528	. . . . . . . . 9 |- F/_ x seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) , CC )
36	35, 9	nffv 5216	. . . . . . . 8 |- F/_ x ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) , CC ) ` m )
37	36	nfeq2 2231	. . . . . . 7 |- F/ x z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) , CC ) ` m )
38	28, 37	nfan 1498	. . . . . 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 1569	. . . . 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 2404	. . . 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 1507	. . 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 4901	. 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 2217	1 |- F/_ x sum_ k e. A B

Syntax hints:   /\ wa 102   \/ wo 662  DECID wdc 776   /\ w3a 920   = wceq 1285   E.wex 1422   e. wcel 1434   F/_wnfc 2207   A.wral 2349   E.wrex 2350   [_csb 2909   C_ wss 2974   ifcif 3359   class class class wbr 3793   |-> cmpt 3847   iotacio 4895   -1-1-onto->wf1o 4931   ` cfv 4932  (class class class)co 5543   CCcc 7041   0cc0 7043   1c1 7044   + caddc 7046   <_ cle 7216   NNcn 8106   ZZcz 8432   ZZ>=cuz 8700   ...cfz 9105   seqcseq 9521   ~~> cli 10255   sum_csu 10328

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-if 3360  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-recs 5954  df-frec 6040  df-iseq 9522  df-isum 10329

This theorem is referenced by: (None)