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Theorem nfsum 12164
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 12159 . 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 2419 . . . . 5  |-  F/_ x ZZ
3 nfsum.1 . . . . . . 7  |-  F/_ x A
4 nfcv 2419 . . . . . . 7  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 3173 . . . . . 6  |-  F/ x  A  C_  ( ZZ>= `  m
)
6 nfcv 2419 . . . . . . . 8  |-  F/_ x m
7 nfcv 2419 . . . . . . . 8  |-  F/_ x  +
83nfcri 2413 . . . . . . . . . 10  |-  F/ x  k  e.  A
9 nfsum.2 . . . . . . . . . 10  |-  F/_ x B
10 nfcv 2419 . . . . . . . . . 10  |-  F/_ x
0
118, 9, 10nfif 3589 . . . . . . . . 9  |-  F/_ x if ( k  e.  A ,  B ,  0 )
122, 11nfmpt 4108 . . . . . . . 8  |-  F/_ x
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
136, 7, 12nfseq 11056 . . . . . . 7  |-  F/_ x  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )
14 nfcv 2419 . . . . . . 7  |-  F/_ x  ~~>
15 nfcv 2419 . . . . . . 7  |-  F/_ x
z
1613, 14, 15nfbr 4067 . . . . . 6  |-  F/ x  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  z
175, 16nfan 1771 . . . . 5  |-  F/ x
( A  C_  ( ZZ>=
`  m )  /\  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  z )
182, 17nfrex 2598 . . . 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 2419 . . . . 5  |-  F/_ x NN
20 nfcv 2419 . . . . . . . 8  |-  F/_ x
f
21 nfcv 2419 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
2220, 21, 3nff1o 5470 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
23 nfcv 2419 . . . . . . . . . 10  |-  F/_ x
1
24 nfcv 2419 . . . . . . . . . . . 12  |-  F/_ x
( f `  n
)
2524, 9nfcsb 3115 . . . . . . . . . . 11  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
2619, 25nfmpt 4108 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
2723, 7, 26nfseq 11056 . . . . . . . . 9  |-  F/_ x  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
2827, 6nffv 5532 . . . . . . . 8  |-  F/_ x
(  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
)
2928nfeq2 2430 . . . . . . 7  |-  F/ x  z  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
3022, 29nfan 1771 . . . . . 6  |-  F/ x
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3130nfex 1767 . . . . 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 2598 . . . 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 1770 . . 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 5223 . 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 2416 1  |-  F/_ x sum_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   F/_wnfc 2406   E.wrex 2544   [_csb 3081    C_ wss 3152   ifcif 3565   class class class wbr 4023    e. cmpt 4077   iotacio 5217   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740   NNcn 9746   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046    ~~> cli 11958   sum_csu 12158
This theorem is referenced by:  fsum2dlem  12233  fsumcom2  12237  fsumrlim  12269  fsumiun  12279  fsumcn  18374  fsum2cn  18375  nfitg1  19128  nfitg  19129  dvmptfsum  19322  fsumdvdscom  20425  fsumcnf  27692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-seq 11047  df-sum 12159
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