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Theorem nfsum 12180
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 12175 . 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 2432 . . . . 5  |-  F/_ x ZZ
3 nfsum.1 . . . . . . 7  |-  F/_ x A
4 nfcv 2432 . . . . . . 7  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 3186 . . . . . 6  |-  F/ x  A  C_  ( ZZ>= `  m
)
6 nfcv 2432 . . . . . . . 8  |-  F/_ x m
7 nfcv 2432 . . . . . . . 8  |-  F/_ x  +
83nfcri 2426 . . . . . . . . . 10  |-  F/ x  k  e.  A
9 nfsum.2 . . . . . . . . . 10  |-  F/_ x B
10 nfcv 2432 . . . . . . . . . 10  |-  F/_ x
0
118, 9, 10nfif 3602 . . . . . . . . 9  |-  F/_ x if ( k  e.  A ,  B ,  0 )
122, 11nfmpt 4124 . . . . . . . 8  |-  F/_ x
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
136, 7, 12nfseq 11072 . . . . . . 7  |-  F/_ x  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )
14 nfcv 2432 . . . . . . 7  |-  F/_ x  ~~>
15 nfcv 2432 . . . . . . 7  |-  F/_ x
z
1613, 14, 15nfbr 4083 . . . . . 6  |-  F/ x  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  z
175, 16nfan 1783 . . . . 5  |-  F/ x
( A  C_  ( ZZ>=
`  m )  /\  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  z )
182, 17nfrex 2611 . . . 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 2432 . . . . 5  |-  F/_ x NN
20 nfcv 2432 . . . . . . . 8  |-  F/_ x
f
21 nfcv 2432 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
2220, 21, 3nff1o 5486 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
23 nfcv 2432 . . . . . . . . . 10  |-  F/_ x
1
24 nfcv 2432 . . . . . . . . . . . 12  |-  F/_ x
( f `  n
)
2524, 9nfcsb 3128 . . . . . . . . . . 11  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
2619, 25nfmpt 4124 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
2723, 7, 26nfseq 11072 . . . . . . . . 9  |-  F/_ x  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
2827, 6nffv 5548 . . . . . . . 8  |-  F/_ x
(  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
)
2928nfeq2 2443 . . . . . . 7  |-  F/ x  z  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
3022, 29nfan 1783 . . . . . 6  |-  F/ x
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3130nfex 1779 . . . . 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 2611 . . . 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 1782 . . 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 5239 . 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 2429 1  |-  F/_ x sum_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   F/_wnfc 2419   E.wrex 2557   [_csb 3094    C_ wss 3165   ifcif 3578   class class class wbr 4039    e. cmpt 4093   iotacio 5233   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756   NNcn 9762   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062    ~~> cli 11974   sum_csu 12174
This theorem is referenced by:  fsum2dlem  12249  fsumcom2  12253  fsumrlim  12285  fsumiun  12295  fsumcn  18390  fsum2cn  18391  nfitg1  19144  nfitg  19145  dvmptfsum  19338  fsumdvdscom  20441  fsumcnf  27795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-seq 11063  df-sum 12175
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