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Theorem nfsum 13487
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  m  n  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sum 13483 . 2  |-  sum_ k  e.  A  B  =  ( iota z ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  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  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
2 nfcv 2603 . . . . 5  |-  F/_ x ZZ
3 nfsum.1 . . . . . . 7  |-  F/_ x A
4 nfcv 2603 . . . . . . 7  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 3479 . . . . . 6  |-  F/ x  A  C_  ( ZZ>= `  m
)
6 nfcv 2603 . . . . . . . 8  |-  F/_ x m
7 nfcv 2603 . . . . . . . 8  |-  F/_ x  +
83nfcri 2596 . . . . . . . . . 10  |-  F/ x  n  e.  A
9 nfcv 2603 . . . . . . . . . . 11  |-  F/_ x n
10 nfsum.2 . . . . . . . . . . 11  |-  F/_ x B
119, 10nfcsb 3435 . . . . . . . . . 10  |-  F/_ x [_ n  /  k ]_ B
12 nfcv 2603 . . . . . . . . . 10  |-  F/_ x
0
138, 11, 12nfif 3951 . . . . . . . . 9  |-  F/_ x if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
142, 13nfmpt 4521 . . . . . . . 8  |-  F/_ x
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) )
156, 7, 14nfseq 12091 . . . . . . 7  |-  F/_ x  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )
16 nfcv 2603 . . . . . . 7  |-  F/_ x  ~~>
17 nfcv 2603 . . . . . . 7  |-  F/_ x
z
1815, 16, 17nfbr 4477 . . . . . 6  |-  F/ x  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  z
195, 18nfan 1912 . . . . 5  |-  F/ x
( A  C_  ( ZZ>=
`  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  z )
202, 19nfrex 2904 . . . 4  |-  F/ x E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  z )
21 nfcv 2603 . . . . 5  |-  F/_ x NN
22 nfcv 2603 . . . . . . . 8  |-  F/_ x
f
23 nfcv 2603 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
2422, 23, 3nff1o 5800 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
25 nfcv 2603 . . . . . . . . . 10  |-  F/_ x
1
26 nfcv 2603 . . . . . . . . . . . 12  |-  F/_ x
( f `  n
)
2726, 10nfcsb 3435 . . . . . . . . . . 11  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
2821, 27nfmpt 4521 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
2925, 7, 28nfseq 12091 . . . . . . . . 9  |-  F/_ x  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
3029, 6nffv 5859 . . . . . . . 8  |-  F/_ x
(  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m )
3130nfeq2 2620 . . . . . . 7  |-  F/ x  z  =  (  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
3224, 31nfan 1912 . . . . . 6  |-  F/ x
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3332nfex 1932 . . . . 5  |-  F/ x E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  z  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3421, 33nfrex 2904 . . . 4  |-  F/ x E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  z  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3520, 34nfor 1919 . . 3  |-  F/ x
( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  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  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) ) )
3635nfiota 5541 . 2  |-  F/_ x
( iota z ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  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  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
371, 36nfcxfr 2601 1  |-  F/_ x 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:  fsum2dlem  13559  fsumcom2  13563  fsumrlim  13599  fsumiun  13609  fsumcn  21240  fsum2cn  21241  nfitg1  22046  nfitg  22047  dvmptfsum  22242  fsumdvdscom  23326  fsumcnf  31343
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