Theorem nfsum1 11521

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 11519	. 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 2339	. . . . 5 |- F/_ k ZZ
3		nfsum1.1	. . . . . . 7 |- F/_ k A
4		nfcv 2339	. . . . . . 7 |- F/_ k ( ZZ>= ` m )
5	3, 4	nfss 3176	. . . . . 6 |- F/ k A C_ ( ZZ>= ` m )
6	3	nfcri 2333	. . . . . . . 8 |- F/ k j e. A
7	6	nfdc 1673	. . . . . . 7 |- F/ kDECID j e. A
8	4, 7	nfralxy 2535	. . . . . 6 |- F/ k A. j e. ( ZZ>= ` m )DECID j e. A
9		nfcv 2339	. . . . . . . 8 |- F/_ k m
10		nfcv 2339	. . . . . . . 8 |- F/_ k +
11	3	nfcri 2333	. . . . . . . . . 10 |- F/ k n e. A
12		nfcsb1v 3117	. . . . . . . . . 10 |- F/_ k [_ n / k ]_ B
13		nfcv 2339	. . . . . . . . . 10 |- F/_ k 0
14	11, 12, 13	nfif 3589	. . . . . . . . 9 |- F/_ k if ( n e. A , [_ n / k ]_ B , 0 )
15	2, 14	nfmpt 4125	. . . . . . . 8 |- F/_ k ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
16	9, 10, 15	nfseq 10549	. . . . . . 7 |- F/_ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
17		nfcv 2339	. . . . . . 7 |- F/_ k ~~>
18		nfcv 2339	. . . . . . 7 |- F/_ k x
19	16, 17, 18	nfbr 4079	. . . . . 6 |- F/ k seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x
20	5, 8, 19	nf3an 1580	. . . . 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 2538	. . . 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 2339	. . . . 5 |- F/_ k NN
23		nfcv 2339	. . . . . . . 8 |- F/_ k f
24		nfcv 2339	. . . . . . . 8 |- F/_ k ( 1 ... m )
25	23, 24, 3	nff1o 5502	. . . . . . 7 |- F/ k f : ( 1 ... m ) -1-1-onto-> A
26		nfcv 2339	. . . . . . . . . 10 |- F/_ k 1
27		nfv 1542	. . . . . . . . . . . 12 |- F/ k n <_ m
28		nfcsb1v 3117	. . . . . . . . . . . 12 |- F/_ k [_ ( f ` n ) / k ]_ B
29	27, 28, 13	nfif 3589	. . . . . . . . . . 11 |- F/_ k if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
30	22, 29	nfmpt 4125	. . . . . . . . . 10 |- F/_ k ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
31	26, 10, 30	nfseq 10549	. . . . . . . . 9 |- F/_ k seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) )
32	31, 9	nffv 5568	. . . . . . . 8 |- F/_ k ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
33	32	nfeq2 2351	. . . . . . 7 |- F/ k x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
34	25, 33	nfan 1579	. . . . . 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 1651	. . . . 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 2538	. . . 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 1588	. . 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 5223	. 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 2336	1 |- F/_ k sum_ k e. A B

Syntax hints:   /\ wa 104   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   E.wex 1506   e. wcel 2167   F/_wnfc 2326   A.wral 2475   E.wrex 2476   [_csb 3084   C_ wss 3157   ifcif 3561   class class class wbr 4033   |-> cmpt 4094   iotacio 5217   -1-1-onto->wf1o 5257   ` cfv 5258  (class class class)co 5922   0cc0 7879   1c1 7880   + caddc 7882   <_ cle 8062   NNcn 8990   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083   seqcseq 10539   ~~> cli 11443   sum_csu 11518

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-if 3562  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-recs 6363  df-frec 6449  df-seqfrec 10540  df-sumdc 11519

This theorem is referenced by:  mertenslem2  11701