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Theorem nfsum1 13486
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  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sum 13483 . 2  |-  sum_ k  e.  A  B  =  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
2 nfcv 2603 . . . . 5  |-  F/_ k ZZ
3 nfsum1.1 . . . . . . 7  |-  F/_ k A
4 nfcv 2603 . . . . . . 7  |-  F/_ k
( ZZ>= `  m )
53, 4nfss 3479 . . . . . 6  |-  F/ k  A  C_  ( ZZ>= `  m )
6 nfcv 2603 . . . . . . . 8  |-  F/_ k
m
7 nfcv 2603 . . . . . . . 8  |-  F/_ k  +
83nfcri 2596 . . . . . . . . . 10  |-  F/ k  n  e.  A
9 nfcsb1v 3433 . . . . . . . . . 10  |-  F/_ k [_ n  /  k ]_ B
10 nfcv 2603 . . . . . . . . . 10  |-  F/_ k
0
118, 9, 10nfif 3951 . . . . . . . . 9  |-  F/_ k if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
122, 11nfmpt 4521 . . . . . . . 8  |-  F/_ k
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) )
136, 7, 12nfseq 12091 . . . . . . 7  |-  F/_ k  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )
14 nfcv 2603 . . . . . . 7  |-  F/_ k  ~~>
15 nfcv 2603 . . . . . . 7  |-  F/_ k
x
1613, 14, 15nfbr 4477 . . . . . 6  |-  F/ k  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x
175, 16nfan 1912 . . . . 5  |-  F/ k ( A  C_  ( ZZ>=
`  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )
182, 17nfrex 2904 . . . 4  |-  F/ k E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )
19 nfcv 2603 . . . . 5  |-  F/_ k NN
20 nfcv 2603 . . . . . . . 8  |-  F/_ k
f
21 nfcv 2603 . . . . . . . 8  |-  F/_ k
( 1 ... m
)
2220, 21, 3nff1o 5800 . . . . . . 7  |-  F/ k  f : ( 1 ... m ) -1-1-onto-> A
23 nfcv 2603 . . . . . . . . . 10  |-  F/_ k
1
24 nfcsb1v 3433 . . . . . . . . . . 11  |-  F/_ k [_ ( f `  n
)  /  k ]_ B
2519, 24nfmpt 4521 . . . . . . . . . 10  |-  F/_ k
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
2623, 7, 25nfseq 12091 . . . . . . . . 9  |-  F/_ k  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
2726, 6nffv 5859 . . . . . . . 8  |-  F/_ k
(  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m )
2827nfeq2 2620 . . . . . . 7  |-  F/ k  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
2922, 28nfan 1912 . . . . . 6  |-  F/ k ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3029nfex 1932 . . . . 5  |-  F/ k E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
) )
3119, 30nfrex 2904 . . . 4  |-  F/ k E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3218, 31nfor 1919 . . 3  |-  F/ k ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )  \/ 
E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) ) )
3332nfiota 5541 . 2  |-  F/_ k
( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
341, 33nfcxfr 2601 1  |-  F/_ k sum_ k  e.  A  B
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    = wceq 1381   E.wex 1597    e. wcel 1802   F/_wnfc 2589   E.wrex 2792   [_csb 3417    C_ wss 3458   ifcif 3922   class class class wbr 4433    |-> cmpt 4491   iotacio 5535   -1-1-onto->wf1o 5573   ` cfv 5574  (class class class)co 6277   0cc0 9490   1c1 9491    + caddc 9493   NNcn 10537   ZZcz 10865   ZZ>=cuz 11085   ...cfz 11676    seqcseq 12081    ~~> cli 13281   sum_csu 13482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-recs 7040  df-rdg 7074  df-seq 12082  df-sum 13483
This theorem is referenced by:  fourierdlem112  31886
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