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Theorem nfsum1 13463
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 13460 . 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 2624 . . . . 5  |-  F/_ k ZZ
3 nfsum1.1 . . . . . . 7  |-  F/_ k A
4 nfcv 2624 . . . . . . 7  |-  F/_ k
( ZZ>= `  m )
53, 4nfss 3492 . . . . . 6  |-  F/ k  A  C_  ( ZZ>= `  m )
6 nfcv 2624 . . . . . . . 8  |-  F/_ k
m
7 nfcv 2624 . . . . . . . 8  |-  F/_ k  +
83nfel2 2642 . . . . . . . . . 10  |-  F/ k  n  e.  A
9 nfcsb1v 3446 . . . . . . . . . 10  |-  F/_ k [_ n  /  k ]_ B
10 nfcv 2624 . . . . . . . . . 10  |-  F/_ k
0
118, 9, 10nfif 3963 . . . . . . . . 9  |-  F/_ k if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
122, 11nfmpt 4530 . . . . . . . 8  |-  F/_ k
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) )
136, 7, 12nfseq 12075 . . . . . . 7  |-  F/_ k  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )
14 nfcv 2624 . . . . . . 7  |-  F/_ k  ~~>
15 nfcv 2624 . . . . . . 7  |-  F/_ k
x
1613, 14, 15nfbr 4486 . . . . . 6  |-  F/ k  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x
175, 16nfan 1870 . . . . 5  |-  F/ k ( A  C_  ( ZZ>=
`  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )
182, 17nfrex 2922 . . . 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 2624 . . . . 5  |-  F/_ k NN
20 nfcv 2624 . . . . . . . 8  |-  F/_ k
f
21 nfcv 2624 . . . . . . . 8  |-  F/_ k
( 1 ... m
)
2220, 21, 3nff1o 5807 . . . . . . 7  |-  F/ k  f : ( 1 ... m ) -1-1-onto-> A
23 nfcv 2624 . . . . . . . . . 10  |-  F/_ k
1
24 nfcsb1v 3446 . . . . . . . . . . 11  |-  F/_ k [_ ( f `  n
)  /  k ]_ B
2519, 24nfmpt 4530 . . . . . . . . . 10  |-  F/_ k
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
2623, 7, 25nfseq 12075 . . . . . . . . 9  |-  F/_ k  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
2726, 6nffv 5866 . . . . . . . 8  |-  F/_ k
(  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m )
2827nfeq2 2641 . . . . . . 7  |-  F/ k  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
2922, 28nfan 1870 . . . . . 6  |-  F/ k ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3029nfex 1890 . . . . 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 2922 . . . 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 1877 . . 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 5548 . 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 2622 1  |-  F/_ k sum_ k  e.  A  B
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762   F/_wnfc 2610   E.wrex 2810   [_csb 3430    C_ wss 3471   ifcif 3934   class class class wbr 4442    |-> cmpt 4500   iotacio 5542   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277   0cc0 9483   1c1 9484    + caddc 9486   NNcn 10527   ZZcz 10855   ZZ>=cuz 11073   ...cfz 11663    seqcseq 12065    ~~> cli 13258   sum_csu 13459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-recs 7034  df-rdg 7068  df-seq 12066  df-sum 13460
This theorem is referenced by:  fourierdlem112  31476
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