Theorem nfsum 11668

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 11665	. 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 2348	. . . . 5 |- F/_ x ZZ
3		nfsum.1	. . . . . . 7 |- F/_ x A
4		nfcv 2348	. . . . . . 7 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 3186	. . . . . 6 |- F/ x A C_ ( ZZ>= ` m )
6	3	nfcri 2342	. . . . . . . 8 |- F/ x j e. A
7	6	nfdc 1682	. . . . . . 7 |- F/ xDECID j e. A
8	4, 7	nfralxy 2544	. . . . . 6 |- F/ x A. j e. ( ZZ>= ` m )DECID j e. A
9		nfcv 2348	. . . . . . . 8 |- F/_ x m
10		nfcv 2348	. . . . . . . 8 |- F/_ x +
11	3	nfcri 2342	. . . . . . . . . 10 |- F/ x n e. A
12		nfcv 2348	. . . . . . . . . . 11 |- F/_ x n
13		nfsum.2	. . . . . . . . . . 11 |- F/_ x B
14	12, 13	nfcsb 3131	. . . . . . . . . 10 |- F/_ x [_ n / k ]_ B
15		nfcv 2348	. . . . . . . . . 10 |- F/_ x 0
16	11, 14, 15	nfif 3599	. . . . . . . . 9 |- F/_ x if ( n e. A , [_ n / k ]_ B , 0 )
17	2, 16	nfmpt 4136	. . . . . . . 8 |- F/_ x ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
18	9, 10, 17	nfseq 10602	. . . . . . 7 |- F/_ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
19		nfcv 2348	. . . . . . 7 |- F/_ x ~~>
20		nfcv 2348	. . . . . . 7 |- F/_ x z
21	18, 19, 20	nfbr 4090	. . . . . 6 |- F/ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> z
22	5, 8, 21	nf3an 1589	. . . . 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 2545	. . . 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 2348	. . . . 5 |- F/_ x NN
25		nfcv 2348	. . . . . . . 8 |- F/_ x f
26		nfcv 2348	. . . . . . . 8 |- F/_ x ( 1 ... m )
27	25, 26, 3	nff1o 5520	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
28		nfcv 2348	. . . . . . . . . 10 |- F/_ x 1
29		nfv 1551	. . . . . . . . . . . 12 |- F/ x n <_ m
30		nfcv 2348	. . . . . . . . . . . . 13 |- F/_ x ( f ` n )
31	30, 13	nfcsb 3131	. . . . . . . . . . . 12 |- F/_ x [_ ( f ` n ) / k ]_ B
32	29, 31, 15	nfif 3599	. . . . . . . . . . 11 |- F/_ x if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
33	24, 32	nfmpt 4136	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
34	28, 10, 33	nfseq 10602	. . . . . . . . 9 |- F/_ x seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) )
35	34, 9	nffv 5586	. . . . . . . 8 |- F/_ x ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
36	35	nfeq2 2360	. . . . . . 7 |- F/ x z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
37	27, 36	nfan 1588	. . . . . 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 1660	. . . . 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 2545	. . . 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 1597	. . 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 5236	. 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 2345	1 |- F/_ x sum_ k e. A B

Syntax hints:   /\ wa 104   \/ wo 710  DECID wdc 836   /\ w3a 981   = wceq 1373   E.wex 1515   e. wcel 2176   F/_wnfc 2335   A.wral 2484   E.wrex 2485   [_csb 3093   C_ wss 3166   ifcif 3571   class class class wbr 4044   |-> cmpt 4105   iotacio 5230   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5944   0cc0 7925   1c1 7926   + caddc 7928   <_ cle 8108   NNcn 9036   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130   seqcseq 10592   ~~> cli 11589   sum_csu 11664

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-if 3572  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-recs 6391  df-frec 6477  df-seqfrec 10593  df-sumdc 11665

This theorem is referenced by:  fsum2dlemstep  11745  fisumcom2  11749  fsumiun  11788  fsumcncntop  15039  dvmptfsum  15197