Theorem nfsum 9876

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 9873	. 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 2178	. . . . 5 |- F/_ x ZZ
3		nfsum.1	. . . . . . 7 |- F/_ x A
4		nfcv 2178	. . . . . . 7 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 2938	. . . . . 6 |- F/ x A C_ ( ZZ>= ` m )
6		nfcv 2178	. . . . . . . 8 |- F/_ x m
7		nfcv 2178	. . . . . . . 8 |- F/_ x +
8	3	nfcri 2172	. . . . . . . . . 10 |- F/ x n e. A
9		nfcv 2178	. . . . . . . . . . 11 |- F/_ x n
10		nfsum.2	. . . . . . . . . . 11 |- F/_ x B
11	9, 10	nfcsb 2884	. . . . . . . . . 10 |- F/_ x [_ n / k ]_ B
12		nfcv 2178	. . . . . . . . . 10 |- F/_ x 0
13	8, 11, 12	nfif 3356	. . . . . . . . 9 |- F/_ x if ( n e. A , [_ n / k ]_ B , 0 )
14	2, 13	nfmpt 3849	. . . . . . . 8 |- F/_ x ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
15		nfcv 2178	. . . . . . . 8 |- F/_ x CC
16	6, 7, 14, 15	nfiseq 9218	. . . . . . 7 |- F/_ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC )
17		nfcv 2178	. . . . . . 7 |- F/_ x ~~>
18		nfcv 2178	. . . . . . 7 |- F/_ x z
19	16, 17, 18	nfbr 3808	. . . . . 6 |- F/ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> z
20	5, 19	nfan 1457	. . . . 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 2363	. . . 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 2178	. . . . 5 |- F/_ x NN
23		nfcv 2178	. . . . . . . 8 |- F/_ x f
24		nfcv 2178	. . . . . . . 8 |- F/_ x ( 1 ... m )
25	23, 24, 3	nff1o 5124	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
26		nfcv 2178	. . . . . . . . . 10 |- F/_ x 1
27		nfcv 2178	. . . . . . . . . . . 12 |- F/_ x ( f ` n )
28	27, 10	nfcsb 2884	. . . . . . . . . . 11 |- F/_ x [_ ( f ` n ) / k ]_ B
29	22, 28	nfmpt 3849	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> [_ ( f ` n ) / k ]_ B )
30	26, 7, 29, 15	nfiseq 9218	. . . . . . . . 9 |- F/_ x seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC )
31	30, 6	nffv 5185	. . . . . . . 8 |- F/_ x ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m )
32	31	nfeq2 2189	. . . . . . 7 |- F/ x z = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m )
33	25, 32	nfan 1457	. . . . . 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 1528	. . . . 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 2363	. . . 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 1466	. . 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 4871	. 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 2175	1 |- F/_ x sum_ k e. A B

Syntax hints:   /\ wa 97   \/ wo 629   = wceq 1243   E.wex 1381   e. wcel 1393   F/_wnfc 2165   E.wrex 2307   [_csb 2852   C_ wss 2917   ifcif 3331   class class class wbr 3764   |-> cmpt 3818   iotacio 4865   -1-1-onto->wf1o 4901   ` cfv 4902  (class class class)co 5512   CCcc 6887   0cc0 6889   1c1 6890   + caddc 6892   NNcn 7914   ZZcz 8245   ZZ>=cuz 8473   ...cfz 8874   seqcseq 9211   ~~> cli 9799   sum_csu 9872

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-if 3332  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-recs 5920  df-frec 5978  df-iseq 9212  df-sum 9873

This theorem is referenced by: (None)