Theorem nfsum1 11319

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

Step	Hyp	Ref	Expression
1		df-sumdc 11317	. 2 |- sum_ k e. A B = ( iota 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 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) ) )
2		nfcv 2312	. . . . 5 |- F/_ k ZZ
3		nfsum1.1	. . . . . . 7 |- F/_ k A
4		nfcv 2312	. . . . . . 7 |- F/_ k ( ZZ>= ` m )
5	3, 4	nfss 3140	. . . . . 6 |- F/ k A C_ ( ZZ>= ` m )
6	3	nfcri 2306	. . . . . . . 8 |- F/ k j e. A
7	6	nfdc 1652	. . . . . . 7 |- F/ kDECID j e. A
8	4, 7	nfralxy 2508	. . . . . 6 |- F/ k A. j e. ( ZZ>= ` m )DECID j e. A
9		nfcv 2312	. . . . . . . 8 |- F/_ k m
10		nfcv 2312	. . . . . . . 8 |- F/_ k +
11	3	nfcri 2306	. . . . . . . . . 10 |- F/ k n e. A
12		nfcsb1v 3082	. . . . . . . . . 10 |- F/_ k [_ n / k ]_ B
13		nfcv 2312	. . . . . . . . . 10 |- F/_ k 0
14	11, 12, 13	nfif 3554	. . . . . . . . 9 |- F/_ k if ( n e. A , [_ n / k ]_ B , 0 )
15	2, 14	nfmpt 4081	. . . . . . . 8 |- F/_ k ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
16	9, 10, 15	nfseq 10411	. . . . . . 7 |- F/_ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
17		nfcv 2312	. . . . . . 7 |- F/_ k ~~>
18		nfcv 2312	. . . . . . 7 |- F/_ k x
19	16, 17, 18	nfbr 4035	. . . . . 6 |- F/ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x
20	5, 8, 19	nf3an 1559	. . . . 5 |- F/ k ( 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 ) ) ) ~~> x )
21	2, 20	nfrexya 2511	. . . 4 |- F/ k 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 ) ) ) ~~> x )
22		nfcv 2312	. . . . 5 |- F/_ k NN
23		nfcv 2312	. . . . . . . 8 |- F/_ k f
24		nfcv 2312	. . . . . . . 8 |- F/_ k ( 1 ... m )
25	23, 24, 3	nff1o 5440	. . . . . . 7 |- F/ k f : ( 1 ... m ) -1-1-onto-> A
26		nfcv 2312	. . . . . . . . . 10 |- F/_ k 1
27		nfv 1521	. . . . . . . . . . . 12 |- F/ k n <_ m
28		nfcsb1v 3082	. . . . . . . . . . . 12 |- F/_ k [_ ( f ` n ) / k ]_ B
29	27, 28, 13	nfif 3554	. . . . . . . . . . 11 |- F/_ k if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
30	22, 29	nfmpt 4081	. . . . . . . . . 10 |- F/_ k ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
31	26, 10, 30	nfseq 10411	. . . . . . . . 9 |- F/_ k seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) )
32	31, 9	nffv 5506	. . . . . . . 8 |- F/_ k ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
33	32	nfeq2 2324	. . . . . . 7 |- F/ k x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
34	25, 33	nfan 1558	. . . . . 6 |- F/ k ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
35	34	nfex 1630	. . . . 5 |- F/ k E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
36	22, 35	nfrexya 2511	. . . 4 |- F/ k E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
37	21, 36	nfor 1567	. . 3 |- F/ k ( 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 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) )
38	37	nfiotaw 5164	. 2 |- F/_ k ( iota 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 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) ) )
39	1, 38	nfcxfr 2309	1 |- F/_ k sum_ k e. A B

Syntax hints:   /\ wa 103   \/ wo 703  DECID wdc 829   /\ w3a 973   = wceq 1348   E.wex 1485   e. wcel 2141   F/_wnfc 2299   A.wral 2448   E.wrex 2449   [_csb 3049   C_ wss 3121   ifcif 3526   class class class wbr 3989   |-> cmpt 4050   iotacio 5158   -1-1-onto->wf1o 5197   ` cfv 5198  (class class class)co 5853   0cc0 7774   1c1 7775   + caddc 7777   <_ cle 7955   NNcn 8878   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   seqcseq 10401   ~~> cli 11241   sum_csu 11316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-recs 6284  df-frec 6370  df-seqfrec 10402  df-sumdc 11317

This theorem is referenced by:  mertenslem2  11499