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Theorem nfsum 12487
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-Jul-2013.)
Hypotheses
Ref Expression
nfsum.1  |-  F/_ x A
nfsum.2  |-  F/_ x B
Assertion
Ref Expression
nfsum  |-  F/_ x sum_ k  e.  A  B
Distinct variable group:    x, k
Allowed substitution hints:    A( x, k)    B( x, k)

Proof of Theorem nfsum
Dummy variables  f  m  n  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sum 12482 . 2  |-  sum_ k  e.  A  B  =  ( iota z ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  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 2574 . . . . 5  |-  F/_ x ZZ
3 nfsum.1 . . . . . . 7  |-  F/_ x A
4 nfcv 2574 . . . . . . 7  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 3343 . . . . . 6  |-  F/ x  A  C_  ( ZZ>= `  m
)
6 nfcv 2574 . . . . . . . 8  |-  F/_ x m
7 nfcv 2574 . . . . . . . 8  |-  F/_ x  +
83nfcri 2568 . . . . . . . . . 10  |-  F/ x  k  e.  A
9 nfsum.2 . . . . . . . . . 10  |-  F/_ x B
10 nfcv 2574 . . . . . . . . . 10  |-  F/_ x
0
118, 9, 10nfif 3765 . . . . . . . . 9  |-  F/_ x if ( k  e.  A ,  B ,  0 )
122, 11nfmpt 4299 . . . . . . . 8  |-  F/_ x
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
136, 7, 12nfseq 11335 . . . . . . 7  |-  F/_ x  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )
14 nfcv 2574 . . . . . . 7  |-  F/_ x  ~~>
15 nfcv 2574 . . . . . . 7  |-  F/_ x
z
1613, 14, 15nfbr 4258 . . . . . 6  |-  F/ x  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  z
175, 16nfan 1847 . . . . 5  |-  F/ x
( A  C_  ( ZZ>=
`  m )  /\  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  z )
182, 17nfrex 2763 . . . 4  |-  F/ x E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  z )
19 nfcv 2574 . . . . 5  |-  F/_ x NN
20 nfcv 2574 . . . . . . . 8  |-  F/_ x
f
21 nfcv 2574 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
2220, 21, 3nff1o 5674 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
23 nfcv 2574 . . . . . . . . . 10  |-  F/_ x
1
24 nfcv 2574 . . . . . . . . . . . 12  |-  F/_ x
( f `  n
)
2524, 9nfcsb 3287 . . . . . . . . . . 11  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
2619, 25nfmpt 4299 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
2723, 7, 26nfseq 11335 . . . . . . . . 9  |-  F/_ x  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
2827, 6nffv 5737 . . . . . . . 8  |-  F/_ x
(  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
)
2928nfeq2 2585 . . . . . . 7  |-  F/ x  z  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
3022, 29nfan 1847 . . . . . 6  |-  F/ x
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3130nfex 1866 . . . . 5  |-  F/ x E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  z  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3219, 31nfrex 2763 . . . 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 ) )
3318, 32nfor 1859 . . 3  |-  F/ x
( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  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 ) ) )
3433nfiota 5424 . 2  |-  F/_ x
( iota z ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  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 ) ) ) )
351, 34nfcxfr 2571 1  |-  F/_ x sum_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    \/ wo 359    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   F/_wnfc 2561   E.wrex 2708   [_csb 3253    C_ wss 3322   ifcif 3741   class class class wbr 4214    e. cmpt 4268   iotacio 5418   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993    + caddc 8995   NNcn 10002   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045    seq cseq 11325    ~~> cli 12280   sum_csu 12481
This theorem is referenced by:  fsum2dlem  12556  fsumcom2  12560  fsumrlim  12592  fsumiun  12602  fsumcn  18902  fsum2cn  18903  nfitg1  19667  nfitg  19668  dvmptfsum  19861  fsumdvdscom  20972  fsumcnf  27670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-recs 6635  df-rdg 6670  df-seq 11326  df-sum 12482
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