Theorem nfsum 12123

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 12120	. 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 2392	. . . . 5 |- F/_ x ZZ
3		nfsum.1	. . . . . . 7 |- F/_ x A
4		nfcv 2392	. . . . . . 7 |- F/_ x ( ZZ>= ` m )
5	3, 4	nfss 3241	. . . . . 6 |- F/ x A C_ ( ZZ>= ` m )
6	3	nfcri 2386	. . . . . . . 8 |- F/ x j e. A
7	6	nfdc 1711	. . . . . . 7 |- F/ xDECID j e. A
8	4, 7	nfralxy 2588	. . . . . 6 |- F/ x A. j e. ( ZZ>= ` m )DECID j e. A
9		nfcv 2392	. . . . . . . 8 |- F/_ x m
10		nfcv 2392	. . . . . . . 8 |- F/_ x +
11	3	nfcri 2386	. . . . . . . . . 10 |- F/ x n e. A
12		nfcv 2392	. . . . . . . . . . 11 |- F/_ x n
13		nfsum.2	. . . . . . . . . . 11 |- F/_ x B
14	12, 13	nfcsb 3185	. . . . . . . . . 10 |- F/_ x [_ n / k ]_ B
15		nfcv 2392	. . . . . . . . . 10 |- F/_ x 0
16	11, 14, 15	nfif 3669	. . . . . . . . 9 |- F/_ x if ( n e. A , [_ n / k ]_ B , 0 )
17	2, 16	nfmpt 4223	. . . . . . . 8 |- F/_ x ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
18	9, 10, 17	nfseq 10894	. . . . . . 7 |- F/_ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
19		nfcv 2392	. . . . . . 7 |- F/_ x ~~>
20		nfcv 2392	. . . . . . 7 |- F/_ x z
21	18, 19, 20	nfbr 4177	. . . . . 6 |- F/ x seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> z
22	5, 8, 21	nf3an 1619	. . . . 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 2589	. . . 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 2392	. . . . 5 |- F/_ x NN
25		nfcv 2392	. . . . . . . 8 |- F/_ x f
26		nfcv 2392	. . . . . . . 8 |- F/_ x ( 1 ... m )
27	25, 26, 3	nff1o 5637	. . . . . . 7 |- F/ x f : ( 1 ... m ) -1-1-onto-> A
28		nfcv 2392	. . . . . . . . . 10 |- F/_ x 1
29		nfv 1581	. . . . . . . . . . . 12 |- F/ x n <_ m
30		nfcv 2392	. . . . . . . . . . . . 13 |- F/_ x ( f ` n )
31	30, 13	nfcsb 3185	. . . . . . . . . . . 12 |- F/_ x [_ ( f ` n ) / k ]_ B
32	29, 31, 15	nfif 3669	. . . . . . . . . . 11 |- F/_ x if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
33	24, 32	nfmpt 4223	. . . . . . . . . 10 |- F/_ x ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
34	28, 10, 33	nfseq 10894	. . . . . . . . 9 |- F/_ x seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) )
35	34, 9	nffv 5705	. . . . . . . 8 |- F/_ x ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
36	35	nfeq2 2404	. . . . . . 7 |- F/ x z = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
37	27, 36	nfan 1618	. . . . . 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 1690	. . . . 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 2589	. . . 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 1627	. . 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 5341	. 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 2389	1 |- F/_ x sum_ k e. A B

This proof depends on syntax axioms:   /\ wa 104   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   F/_wnfc 2379   A.wral 2528   E.wrex 2529   [_csb 3147   C_ wss 3220   ifcif 3638   class class class wbr 4130   |-> cmpt 4192   iotacio 5335   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   0cc0 8179   1c1 8180   + caddc 8182   <_ cle 8361   NNcn 9304   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411   seqcseq 10884   ~~> cli 12044   sum_csu 12119

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-recs 6576  df-frec 6662  df-seqfrec 10885  df-sumdc 12120

This theorem is used by:  fsum2dlemstep  12201  fisumcom2  12205  fsumiun  12244  fsumcncntop  15668  dvmptfsum  15826