Theorem nfsum1 10193

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 10191	. 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 2219	. . . . 5 |- F/_ k ZZ
3		nfsum1.1	. . . . . . 7 |- F/_ k A
4		nfcv 2219	. . . . . . 7 |- F/_ k ( ZZ>= ` m )
5	3, 4	nfss 2992	. . . . . 6 |- F/ k A C_ ( ZZ>= ` m )
6		nfcv 2219	. . . . . . . 8 |- F/_ k m
7		nfcv 2219	. . . . . . . 8 |- F/_ k +
8	3	nfcri 2213	. . . . . . . . . 10 |- F/ k n e. A
9		nfcsb1v 2938	. . . . . . . . . 10 |- F/_ k [_ n / k ]_ B
10		nfcv 2219	. . . . . . . . . 10 |- F/_ k 0
11	8, 9, 10	nfif 3377	. . . . . . . . 9 |- F/_ k if ( n e. A , [_ n / k ]_ B , 0 )
12	2, 11	nfmpt 3870	. . . . . . . 8 |- F/_ k ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
13		nfcv 2219	. . . . . . . 8 |- F/_ k CC
14	6, 7, 12, 13	nfiseq 9438	. . . . . . 7 |- F/_ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC )
15		nfcv 2219	. . . . . . 7 |- F/_ k ~~>
16		nfcv 2219	. . . . . . 7 |- F/_ k x
17	14, 15, 16	nfbr 3829	. . . . . 6 |- F/ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) , CC ) ~~> x
18	5, 17	nfan 1497	. . . . 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 2405	. . . 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 2219	. . . . 5 |- F/_ k NN
21		nfcv 2219	. . . . . . . 8 |- F/_ k f
22		nfcv 2219	. . . . . . . 8 |- F/_ k ( 1 ... m )
23	21, 22, 3	nff1o 5144	. . . . . . 7 |- F/ k f : ( 1 ... m ) -1-1-onto-> A
24		nfcv 2219	. . . . . . . . . 10 |- F/_ k 1
25		nfcsb1v 2938	. . . . . . . . . . 11 |- F/_ k [_ ( f ` n ) / k ]_ B
26	20, 25	nfmpt 3870	. . . . . . . . . 10 |- F/_ k ( n e. NN |-> [_ ( f ` n ) / k ]_ B )
27	24, 7, 26, 13	nfiseq 9438	. . . . . . . . 9 |- F/_ k seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC )
28	27, 6	nffv 5205	. . . . . . . 8 |- F/_ k ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m )
29	28	nfeq2 2230	. . . . . . 7 |- F/ k x = ( seq 1 ( + , ( n e. NN |-> [_ ( f ` n ) / k ]_ B ) , CC ) ` m )
30	23, 29	nfan 1497	. . . . . 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 1568	. . . . 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 2405	. . . 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 1506	. . 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 4891	. 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 2216	1 |- F/_ k 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)