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Theorem nfsum1 10808
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.
StepHypRef Expression
1 df-isum 10806 . 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 ) ) ,  CC )  ~~>  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 ) ) ,  CC ) `
 m ) ) ) )
2 nfcv 2229 . . . . 5  |-  F/_ k ZZ
3 nfsum1.1 . . . . . . 7  |-  F/_ k A
4 nfcv 2229 . . . . . . 7  |-  F/_ k
( ZZ>= `  m )
53, 4nfss 3021 . . . . . 6  |-  F/ k  A  C_  ( ZZ>= `  m )
63nfcri 2223 . . . . . . . 8  |-  F/ k  j  e.  A
76nfdc 1595 . . . . . . 7  |-  F/ kDECID  j  e.  A
84, 7nfralxy 2415 . . . . . 6  |-  F/ k A. j  e.  (
ZZ>= `  m )DECID  j  e.  A
9 nfcv 2229 . . . . . . . 8  |-  F/_ k
m
10 nfcv 2229 . . . . . . . 8  |-  F/_ k  +
113nfcri 2223 . . . . . . . . . 10  |-  F/ k  n  e.  A
12 nfcsb1v 2966 . . . . . . . . . 10  |-  F/_ k [_ n  /  k ]_ B
13 nfcv 2229 . . . . . . . . . 10  |-  F/_ k
0
1411, 12, 13nfif 3425 . . . . . . . . 9  |-  F/_ k if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
152, 14nfmpt 3938 . . . . . . . 8  |-  F/_ k
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) )
16 nfcv 2229 . . . . . . . 8  |-  F/_ k CC
179, 10, 15, 16nfiseq 9931 . . . . . . 7  |-  F/_ k  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )
18 nfcv 2229 . . . . . . 7  |-  F/_ k  ~~>
19 nfcv 2229 . . . . . . 7  |-  F/_ k
x
2017, 18, 19nfbr 3897 . . . . . 6  |-  F/ k  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )  ~~>  x
215, 8, 20nf3an 1504 . . . . 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 ) ) ,  CC )  ~~>  x )
222, 21nfrexya 2418 . . . 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 ) ) ,  CC )  ~~>  x )
23 nfcv 2229 . . . . 5  |-  F/_ k NN
24 nfcv 2229 . . . . . . . 8  |-  F/_ k
f
25 nfcv 2229 . . . . . . . 8  |-  F/_ k
( 1 ... m
)
2624, 25, 3nff1o 5266 . . . . . . 7  |-  F/ k  f : ( 1 ... m ) -1-1-onto-> A
27 nfcv 2229 . . . . . . . . . 10  |-  F/_ k
1
28 nfv 1467 . . . . . . . . . . . 12  |-  F/ k  n  <_  m
29 nfcsb1v 2966 . . . . . . . . . . . 12  |-  F/_ k [_ ( f `  n
)  /  k ]_ B
3028, 29, 13nfif 3425 . . . . . . . . . . 11  |-  F/_ k if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 )
3123, 30nfmpt 3938 . . . . . . . . . 10  |-  F/_ k
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )
3227, 10, 31, 16nfiseq 9931 . . . . . . . . 9  |-  F/_ k  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ,  CC )
3332, 9nffv 5330 . . . . . . . 8  |-  F/_ k
(  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m )
3433nfeq2 2241 . . . . . . 7  |-  F/ k  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ,  CC ) `
 m )
3526, 34nfan 1503 . . . . . 6  |-  F/ k ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m ) )
3635nfex 1574 . . . . 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 ) ) ,  CC ) `
 m ) )
3723, 36nfrexya 2418 . . . 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 ) ) ,  CC ) `
 m ) )
3822, 37nfor 1512 . . 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 ) ) ,  CC )  ~~>  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 ) ) ,  CC ) `
 m ) ) )
3938nfiotaxy 4999 . 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 ) ) ,  CC )  ~~>  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 ) ) ,  CC ) `
 m ) ) ) )
401, 39nfcxfr 2226 1  |-  F/_ k 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:  mertenslem2  10993
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