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Theorem nfsum 13164
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 13160 . 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 2577 . . . . 5  |-  F/_ x ZZ
3 nfsum.1 . . . . . . 7  |-  F/_ x A
4 nfcv 2577 . . . . . . 7  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 3346 . . . . . 6  |-  F/ x  A  C_  ( ZZ>= `  m
)
6 nfcv 2577 . . . . . . . 8  |-  F/_ x m
7 nfcv 2577 . . . . . . . 8  |-  F/_ x  +
83nfel2 2589 . . . . . . . . . 10  |-  F/ x  n  e.  A
9 nfcv 2577 . . . . . . . . . . 11  |-  F/_ x n
10 nfsum.2 . . . . . . . . . . 11  |-  F/_ x B
119, 10nfcsb 3303 . . . . . . . . . 10  |-  F/_ x [_ n  /  k ]_ B
12 nfcv 2577 . . . . . . . . . 10  |-  F/_ x
0
138, 11, 12nfif 3815 . . . . . . . . 9  |-  F/_ x if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
142, 13nfmpt 4377 . . . . . . . 8  |-  F/_ x
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) )
156, 7, 14nfseq 11812 . . . . . . 7  |-  F/_ x  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )
16 nfcv 2577 . . . . . . 7  |-  F/_ x  ~~>
17 nfcv 2577 . . . . . . 7  |-  F/_ x
z
1815, 16, 17nfbr 4333 . . . . . 6  |-  F/ x  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  z
195, 18nfan 1865 . . . . 5  |-  F/ x
( A  C_  ( ZZ>=
`  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  z )
202, 19nfrex 2769 . . . 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 2577 . . . . 5  |-  F/_ x NN
22 nfcv 2577 . . . . . . . 8  |-  F/_ x
f
23 nfcv 2577 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
2422, 23, 3nff1o 5636 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
25 nfcv 2577 . . . . . . . . . 10  |-  F/_ x
1
26 nfcv 2577 . . . . . . . . . . . 12  |-  F/_ x
( f `  n
)
2726, 10nfcsb 3303 . . . . . . . . . . 11  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
2821, 27nfmpt 4377 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
2925, 7, 28nfseq 11812 . . . . . . . . 9  |-  F/_ x  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
3029, 6nffv 5695 . . . . . . . 8  |-  F/_ x
(  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m )
3130nfeq2 2588 . . . . . . 7  |-  F/ x  z  =  (  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
3224, 31nfan 1865 . . . . . 6  |-  F/ x
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3332nfex 1878 . . . . 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 2769 . . . 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 1872 . . 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 5382 . 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 2574 1  |-  F/_ x sum_ k  e.  A  B
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    = wceq 1364   E.wex 1591    e. wcel 1761   F/_wnfc 2564   E.wrex 2714   [_csb 3285    C_ wss 3325   ifcif 3788   class class class wbr 4289    e. cmpt 4347   iotacio 5376   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279    + caddc 9281   NNcn 10318   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433    seqcseq 11802    ~~> cli 12958   sum_csu 13159
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 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-recs 6828  df-rdg 6862  df-seq 11803  df-sum 13160
This theorem is referenced by:  fsum2dlem  13233  fsumcom2  13237  fsumrlim  13270  fsumiun  13280  fsumcn  20405  fsum2cn  20406  nfitg1  21210  nfitg  21211  dvmptfsum  21406  fsumdvdscom  22484  fsumcnf  29668
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