Theorem nfsum 11863

Description: Bound-variable hypothesis builder for sum: if x is (effectively) not free in A and B, it is not free in sum_ k e. A B. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Hypotheses

Ref	Expression
nfsum.1	|- F/_ x A
nfsum.2	|- F/_ x B

Assertion

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

Proof of Theorem nfsum

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

Step	Hyp	Ref	Expression
1		df-sumdc 11860	. 2 |- sum_ k e. A B = ( iota z ( 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 ) ) ) ~~> z ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) ) )
2		nfcv 2372	. . . . 5 |- F/_ x ZZ
3		nfsum.1	. . . . . . 7 |- F/_ x A
4		nfcv 2372	. . . . . . 7 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 3217	. . . . . 6 |- F/ x A C_ ( ZZ>= ` m )
6	3	nfcri 2366	. . . . . . . 8 |- F/ x j e. A
7	6	nfdc 1705	. . . . . . 7 |- F/ xDECID j e. A
8	4, 7	nfralxy 2568	. . . . . 6 |- F/ x A. j e. ( ZZ>= ` m )DECID j e. A
9		nfcv 2372	. . . . . . . 8 |- F/_ x m
10		nfcv 2372	. . . . . . . 8 |- F/_ x +
11	3	nfcri 2366	. . . . . . . . . 10 |- F/ x n e. A
12		nfcv 2372	. . . . . . . . . . 11 |- F/_ x n
13		nfsum.2	. . . . . . . . . . 11 |- F/_ x B
14	12, 13	nfcsb 3162	. . . . . . . . . 10 |- F/_ x [_ n / k ]_ B
15		nfcv 2372	. . . . . . . . . 10 |- F/_ x 0
16	11, 14, 15	nfif 3631	. . . . . . . . 9 |- F/_ x if ( n e. A , [_ n / k ]_ B , 0 )
17	2, 16	nfmpt 4175	. . . . . . . 8 |- F/_ x ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
18	9, 10, 17	nfseq 10674	. . . . . . 7 |- F/_ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
19		nfcv 2372	. . . . . . 7 |- F/_ x ~~>
20		nfcv 2372	. . . . . . 7 |- F/_ x z
21	18, 19, 20	nfbr 4129	. . . . . 6 |- F/ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> z
22	5, 8, 21	nf3an 1612	. . . . 5 |- F/ x ( 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 ) ) ) ~~> z )
23	2, 22	nfrexw 2569	. . . 4 |- F/ 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 ) ) ) ~~> z )
24		nfcv 2372	. . . . 5 |- F/_ x NN
25		nfcv 2372	. . . . . . . 8 |- F/_ x f
26		nfcv 2372	. . . . . . . 8 |- F/_ x ( 1 ... m )
27	25, 26, 3	nff1o 5569	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
28		nfcv 2372	. . . . . . . . . 10 |- F/_ x 1
29		nfv 1574	. . . . . . . . . . . 12 |- F/ x n <_ m
30		nfcv 2372	. . . . . . . . . . . . 13 |- F/_ x ( f ` n )
31	30, 13	nfcsb 3162	. . . . . . . . . . . 12 |- F/_ x [_ ( f ` n ) / k ]_ B
32	29, 31, 15	nfif 3631	. . . . . . . . . . 11 |- F/_ x if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
33	24, 32	nfmpt 4175	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
34	28, 10, 33	nfseq 10674	. . . . . . . . 9 |- F/_ x seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) )
35	34, 9	nffv 5636	. . . . . . . 8 |- F/_ x ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
36	35	nfeq2 2384	. . . . . . 7 |- F/ x z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
37	27, 36	nfan 1611	. . . . . 6 |- F/ x ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
38	37	nfex 1683	. . . . 5 |- F/ x E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
39	24, 38	nfrexw 2569	. . . 4 |- F/ x E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
40	23, 39	nfor 1620	. . 3 |- F/ 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 ) ) ) ~~> z ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) )
41	40	nfiotaw 5281	. 2 |- F/_ x ( iota z ( 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 ) ) ) ~~> z ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) ) )
42	1, 41	nfcxfr 2369	1 |- F/_ x 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 4082   |-> cmpt 4144   iotacio 5275   -1-1-onto->wf1o 5316   ` cfv 5317  (class class class)co 6000   0cc0 7995   1c1 7996   + caddc 7998   <_ cle 8178   NNcn 9106   ZZcz 9442   ZZ>=cuz 9718   ...cfz 10200   seqcseq 10664   ~~> cli 11784   sum_csu 11859

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 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-recs 6449  df-frec 6535  df-seqfrec 10665  df-sumdc 11860

This theorem is referenced by:  fsum2dlemstep  11940  fisumcom2  11944  fsumiun  11983  fsumcncntop  15235  dvmptfsum  15393