Theorem nfsum 10194

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

Step	Hyp	Ref	Expression
1		df-sum 10191	. 2 |- sum_ k e. A B = ( iota z ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ 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 |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m ) ) ) )
2		nfcv 2219	. . . . 5 |- F/_ x ZZ
3		nfsum.1	. . . . . . 7 |- F/_ x A
4		nfcv 2219	. . . . . . 7 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 2992	. . . . . 6 |- F/ x A C_ ( ZZ>= ` m )
6		nfcv 2219	. . . . . . . 8 |- F/_ x m
7		nfcv 2219	. . . . . . . 8 |- F/_ x +
8	3	nfcri 2213	. . . . . . . . . 10 |- F/ x n e. A
9		nfcv 2219	. . . . . . . . . . 11 |- F/_ x n
10		nfsum.2	. . . . . . . . . . 11 |- F/_ x B
11	9, 10	nfcsb 2940	. . . . . . . . . 10 |- F/_ x [_ n / k ]_ B
12		nfcv 2219	. . . . . . . . . 10 |- F/_ x 0
13	8, 11, 12	nfif 3377	. . . . . . . . 9 |- F/_ x if ( n e. A , [_ n / k ]_ B , 0 )
14	2, 13	nfmpt 3870	. . . . . . . 8 |- F/_ x ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
15		nfcv 2219	. . . . . . . 8 |- F/_ x CC
16	6, 7, 14, 15	nfiseq 9438	. . . . . . 7 |- F/_ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC )
17		nfcv 2219	. . . . . . 7 |- F/_ x ~~>
18		nfcv 2219	. . . . . . 7 |- F/_ x z
19	16, 17, 18	nfbr 3829	. . . . . 6 |- F/ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> z
20	5, 19	nfan 1497	. . . . 5 |- F/ x ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> z )
21	2, 20	nfrexya 2405	. . . 4 |- F/ x E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> z )
22		nfcv 2219	. . . . 5 |- F/_ x NN
23		nfcv 2219	. . . . . . . 8 |- F/_ x f
24		nfcv 2219	. . . . . . . 8 |- F/_ x ( 1 ... m )
25	23, 24, 3	nff1o 5144	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
26		nfcv 2219	. . . . . . . . . 10 |- F/_ x 1
27		nfcv 2219	. . . . . . . . . . . 12 |- F/_ x ( f ` n )
28	27, 10	nfcsb 2940	. . . . . . . . . . 11 |- F/_ x [_ ( f ` n ) / k ]_ B
29	22, 28	nfmpt 3870	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> [_ ( f ` n ) / k ]_ B )
30	26, 7, 29, 15	nfiseq 9438	. . . . . . . . 9 |- F/_ x seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC )
31	30, 6	nffv 5205	. . . . . . . 8 |- F/_ x ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m )
32	31	nfeq2 2230	. . . . . . 7 |- F/ x z = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m )
33	25, 32	nfan 1497	. . . . . 6 |- F/ x ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m ) )
34	33	nfex 1568	. . . . 5 |- F/ x E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m ) )
35	22, 34	nfrexya 2405	. . . 4 |- F/ x E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m ) )
36	21, 35	nfor 1506	. . 3 |- F/ x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ 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 |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m ) ) )
37	36	nfiotaxy 4891	. 2 |- F/_ x ( iota z ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ 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 |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m ) ) ) )
38	1, 37	nfcxfr 2216	1 |- F/_ x sum_ k e. A B

Syntax hints:   /\ wa 102   \/ wo 661   = wceq 1284   E.wex 1421   e. wcel 1433   F/_wnfc 2206   E.wrex 2349   [_csb 2908   C_ wss 2973   ifcif 3351   class class class wbr 3785   |-> cmpt 3839   iotacio 4885   -1-1-onto->wf1o 4921   ` cfv 4922  (class class class)co 5532   CCcc 6979   0cc0 6981   1c1 6982   + caddc 6984   NNcn 8039   ZZcz 8351   ZZ>=cuz 8619   ...cfz 9029   seqcseq 9431   ~~> cli 10117   sum_csu 10190

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-if 3352  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-recs 5943  df-frec 6001  df-iseq 9432  df-sum 10191

This theorem is referenced by: (None)