Theorem nfsum1 9875

Description: Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Hypothesis

Ref	Expression
nfsum1.1	|- F/_ k A

Assertion

Ref	Expression
nfsum1	|- F/_ k sum_ k e. A B

Proof of Theorem nfsum1

Dummy variables f m n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sum 9873	. 2 |- sum_ k e. A B = ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m ) ) ) )
2		nfcv 2178	. . . . 5 |- F/_ k ZZ
3		nfsum1.1	. . . . . . 7 |- F/_ k A
4		nfcv 2178	. . . . . . 7 |- F/_ k ( ZZ>= ` m )
5	3, 4	nfss 2938	. . . . . 6 |- F/ k A C_ ( ZZ>= ` m )
6		nfcv 2178	. . . . . . . 8 |- F/_ k m
7		nfcv 2178	. . . . . . . 8 |- F/_ k +
8	3	nfcri 2172	. . . . . . . . . 10 |- F/ k n e. A
9		nfcsb1v 2882	. . . . . . . . . 10 |- F/_ k [_ n / k ]_ B
10		nfcv 2178	. . . . . . . . . 10 |- F/_ k 0
11	8, 9, 10	nfif 3356	. . . . . . . . 9 |- F/_ k if ( n e. A , [_ n / k ]_ B , 0 )
12	2, 11	nfmpt 3849	. . . . . . . 8 |- F/_ k ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
13		nfcv 2178	. . . . . . . 8 |- F/_ k CC
14	6, 7, 12, 13	nfiseq 9218	. . . . . . 7 |- F/_ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC )
15		nfcv 2178	. . . . . . 7 |- F/_ k ~~>
16		nfcv 2178	. . . . . . 7 |- F/_ k x
17	14, 15, 16	nfbr 3808	. . . . . 6 |- F/ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> x
18	5, 17	nfan 1457	. . . . 5 |- F/ k ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> x )
19	2, 18	nfrexya 2363	. . . 4 |- F/ k E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> x )
20		nfcv 2178	. . . . 5 |- F/_ k NN
21		nfcv 2178	. . . . . . . 8 |- F/_ k f
22		nfcv 2178	. . . . . . . 8 |- F/_ k ( 1 ... m )
23	21, 22, 3	nff1o 5124	. . . . . . 7 |- F/ k f : ( 1 ... m ) -1-1-onto-> A
24		nfcv 2178	. . . . . . . . . 10 |- F/_ k 1
25		nfcsb1v 2882	. . . . . . . . . . 11 |- F/_ k [_ ( f ` n ) / k ]_ B
26	20, 25	nfmpt 3849	. . . . . . . . . 10 |- F/_ k ( n e. NN |-> [_ ( f ` n ) / k ]_ B )
27	24, 7, 26, 13	nfiseq 9218	. . . . . . . . 9 |- F/_ k seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC )
28	27, 6	nffv 5185	. . . . . . . 8 |- F/_ k ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m )
29	28	nfeq2 2189	. . . . . . 7 |- F/ k x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m )
30	23, 29	nfan 1457	. . . . . 6 |- F/ k ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m ) )
31	30	nfex 1528	. . . . 5 |- F/ k E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m ) )
32	20, 31	nfrexya 2363	. . . 4 |- F/ k E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m ) )
33	19, 32	nfor 1466	. . 3 |- F/ k ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m ) ) )
34	33	nfiotaxy 4871	. 2 |- F/_ k ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m ) ) ) )
35	1, 34	nfcxfr 2175	1 |- F/_ k 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)