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Theorem nfsum 10809
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.
StepHypRef Expression
1 df-isum 10806 . 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 ) ) ,  CC )  ~~>  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 ) ) ,  CC ) `
 m ) ) ) )
2 nfcv 2229 . . . . 5  |-  F/_ x ZZ
3 nfsum.1 . . . . . . 7  |-  F/_ x A
4 nfcv 2229 . . . . . . 7  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 3021 . . . . . 6  |-  F/ x  A  C_  ( ZZ>= `  m
)
63nfcri 2223 . . . . . . . 8  |-  F/ x  j  e.  A
76nfdc 1595 . . . . . . 7  |-  F/ xDECID  j  e.  A
84, 7nfralxy 2415 . . . . . 6  |-  F/ x A. j  e.  ( ZZ>=
`  m )DECID  j  e.  A
9 nfcv 2229 . . . . . . . 8  |-  F/_ x m
10 nfcv 2229 . . . . . . . 8  |-  F/_ x  +
113nfcri 2223 . . . . . . . . . 10  |-  F/ x  n  e.  A
12 nfcv 2229 . . . . . . . . . . 11  |-  F/_ x n
13 nfsum.2 . . . . . . . . . . 11  |-  F/_ x B
1412, 13nfcsb 2968 . . . . . . . . . 10  |-  F/_ x [_ n  /  k ]_ B
15 nfcv 2229 . . . . . . . . . 10  |-  F/_ x
0
1611, 14, 15nfif 3425 . . . . . . . . 9  |-  F/_ x if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
172, 16nfmpt 3938 . . . . . . . 8  |-  F/_ x
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) )
18 nfcv 2229 . . . . . . . 8  |-  F/_ x CC
199, 10, 17, 18nfiseq 9931 . . . . . . 7  |-  F/_ x  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )
20 nfcv 2229 . . . . . . 7  |-  F/_ x  ~~>
21 nfcv 2229 . . . . . . 7  |-  F/_ x
z
2219, 20, 21nfbr 3897 . . . . . 6  |-  F/ x  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )  ~~>  z
235, 8, 22nf3an 1504 . . . . 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 ) ) ,  CC )  ~~>  z )
242, 23nfrexxy 2416 . . . 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 ) ) ,  CC )  ~~>  z )
25 nfcv 2229 . . . . 5  |-  F/_ x NN
26 nfcv 2229 . . . . . . . 8  |-  F/_ x
f
27 nfcv 2229 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
2826, 27, 3nff1o 5266 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
29 nfcv 2229 . . . . . . . . . 10  |-  F/_ x
1
30 nfv 1467 . . . . . . . . . . . 12  |-  F/ x  n  <_  m
31 nfcv 2229 . . . . . . . . . . . . 13  |-  F/_ x
( f `  n
)
3231, 13nfcsb 2968 . . . . . . . . . . . 12  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
3330, 32, 15nfif 3425 . . . . . . . . . . 11  |-  F/_ x if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 )
3425, 33nfmpt 3938 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )
3529, 10, 34, 18nfiseq 9931 . . . . . . . . 9  |-  F/_ x  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ,  CC )
3635, 9nffv 5330 . . . . . . . 8  |-  F/_ x
(  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m )
3736nfeq2 2241 . . . . . . 7  |-  F/ x  z  =  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ,  CC ) `
 m )
3828, 37nfan 1503 . . . . . 6  |-  F/ x
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m ) )
3938nfex 1574 . . . . 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 ) ) ,  CC ) `
 m ) )
4025, 39nfrexxy 2416 . . . 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 ) ) ,  CC ) `
 m ) )
4124, 40nfor 1512 . . 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 ) ) ,  CC )  ~~>  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 ) ) ,  CC ) `
 m ) ) )
4241nfiotaxy 4999 . 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 ) ) ,  CC )  ~~>  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 ) ) ,  CC ) `
 m ) ) ) )
431, 42nfcxfr 2226 1  |-  F/_ x sum_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 665  DECID wdc 781    /\ w3a 925    = wceq 1290   E.wex 1427    e. wcel 1439   F/_wnfc 2216   A.wral 2360   E.wrex 2361   [_csb 2936    C_ wss 3002   ifcif 3399   class class class wbr 3853    |-> cmpt 3907   iotacio 4993   -1-1-onto->wf1o 5029   ` cfv 5030  (class class class)co 5668   CCcc 7411   0cc0 7413   1c1 7414    + caddc 7416    <_ cle 7586   NNcn 8485   ZZcz 8813   ZZ>=cuz 9082   ...cfz 9487    seqcseq4 9914    ~~> cli 10729   sum_csu 10805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-un 3006  df-in 3008  df-ss 3015  df-if 3400  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-recs 6086  df-frec 6172  df-iseq 9916  df-isum 10806
This theorem is referenced by:  fsum2dlemstep  10891  fisumcom2  10895  fsumiun  10934
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