Theorem nfsum1 11867

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 11865	. 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 2372	. . . . 5 |- F/_ k ZZ
3		nfsum1.1	. . . . . . 7 |- F/_ k A
4		nfcv 2372	. . . . . . 7 |- F/_ k ( ZZ>= ` m )
5	3, 4	nfss 3217	. . . . . 6 |- F/ k A C_ ( ZZ>= ` m )
6	3	nfcri 2366	. . . . . . . 8 |- F/ k j e. A
7	6	nfdc 1705	. . . . . . 7 |- F/ kDECID j e. A
8	4, 7	nfralxy 2568	. . . . . 6 |- F/ k A. j e. ( ZZ>= ` m )DECID j e. A
9		nfcv 2372	. . . . . . . 8 |- F/_ k m
10		nfcv 2372	. . . . . . . 8 |- F/_ k +
11	3	nfcri 2366	. . . . . . . . . 10 |- F/ k n e. A
12		nfcsb1v 3157	. . . . . . . . . 10 |- F/_ k [_ n / k ]_ B
13		nfcv 2372	. . . . . . . . . 10 |- F/_ k 0
14	11, 12, 13	nfif 3631	. . . . . . . . 9 |- F/_ k if ( n e. A , [_ n / k ]_ B , 0 )
15	2, 14	nfmpt 4176	. . . . . . . 8 |- F/_ k ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
16	9, 10, 15	nfseq 10679	. . . . . . 7 |- F/_ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
17		nfcv 2372	. . . . . . 7 |- F/_ k ~~>
18		nfcv 2372	. . . . . . 7 |- F/_ k x
19	16, 17, 18	nfbr 4130	. . . . . 6 |- F/ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x
20	5, 8, 19	nf3an 1612	. . . . 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 2571	. . . 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 2372	. . . . 5 |- F/_ k NN
23		nfcv 2372	. . . . . . . 8 |- F/_ k f
24		nfcv 2372	. . . . . . . 8 |- F/_ k ( 1 ... m )
25	23, 24, 3	nff1o 5570	. . . . . . 7 |- F/ k f : ( 1 ... m ) -1-1-onto-> A
26		nfcv 2372	. . . . . . . . . 10 |- F/_ k 1
27		nfv 1574	. . . . . . . . . . . 12 |- F/ k n <_ m
28		nfcsb1v 3157	. . . . . . . . . . . 12 |- F/_ k [_ ( f ` n ) / k ]_ B
29	27, 28, 13	nfif 3631	. . . . . . . . . . 11 |- F/_ k if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
30	22, 29	nfmpt 4176	. . . . . . . . . 10 |- F/_ k ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
31	26, 10, 30	nfseq 10679	. . . . . . . . 9 |- F/_ k seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) )
32	31, 9	nffv 5637	. . . . . . . 8 |- F/_ k ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
33	32	nfeq2 2384	. . . . . . 7 |- F/ k x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
34	25, 33	nfan 1611	. . . . . 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 1683	. . . . 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 2571	. . . 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 1620	. . 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 5282	. 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 2369	1 |- F/_ k sum_ k e. A B

Syntax hints:   /\ wa 104   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   F/_wnfc 2359   A.wral 2508   E.wrex 2509   [_csb 3124   C_ wss 3197   ifcif 3602   class class class wbr 4083   |-> cmpt 4145   iotacio 5276   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6001   0cc0 7999   1c1 8000   + caddc 8002   <_ cle 8182   NNcn 9110   ZZcz 9446   ZZ>=cuz 9722   ...cfz 10204   seqcseq 10669   ~~> cli 11789   sum_csu 11864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-recs 6451  df-frec 6537  df-seqfrec 10670  df-sumdc 11865

This theorem is referenced by:  mertenslem2  12047