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Theorem nfsum 11382
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  j  m  n  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sumdc 11379 . 2  |-  sum_ k  e.  A  B  =  ( iota z ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  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  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ) `  m
) ) ) )
2 nfcv 2331 . . . . 5  |-  F/_ x ZZ
3 nfsum.1 . . . . . . 7  |-  F/_ x A
4 nfcv 2331 . . . . . . 7  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 3162 . . . . . 6  |-  F/ x  A  C_  ( ZZ>= `  m
)
63nfcri 2325 . . . . . . . 8  |-  F/ x  j  e.  A
76nfdc 1669 . . . . . . 7  |-  F/ xDECID  j  e.  A
84, 7nfralxy 2527 . . . . . 6  |-  F/ x A. j  e.  ( ZZ>=
`  m )DECID  j  e.  A
9 nfcv 2331 . . . . . . . 8  |-  F/_ x m
10 nfcv 2331 . . . . . . . 8  |-  F/_ x  +
113nfcri 2325 . . . . . . . . . 10  |-  F/ x  n  e.  A
12 nfcv 2331 . . . . . . . . . . 11  |-  F/_ x n
13 nfsum.2 . . . . . . . . . . 11  |-  F/_ x B
1412, 13nfcsb 3108 . . . . . . . . . 10  |-  F/_ x [_ n  /  k ]_ B
15 nfcv 2331 . . . . . . . . . 10  |-  F/_ x
0
1611, 14, 15nfif 3576 . . . . . . . . 9  |-  F/_ x if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
172, 16nfmpt 4109 . . . . . . . 8  |-  F/_ x
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) )
189, 10, 17nfseq 10472 . . . . . . 7  |-  F/_ x  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )
19 nfcv 2331 . . . . . . 7  |-  F/_ x  ~~>
20 nfcv 2331 . . . . . . 7  |-  F/_ x
z
2118, 19, 20nfbr 4063 . . . . . 6  |-  F/ x  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  z
225, 8, 21nf3an 1576 . . . . 5  |-  F/ x
( A  C_  ( ZZ>=
`  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  z )
232, 22nfrexxy 2528 . . . 4  |-  F/ x E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  z )
24 nfcv 2331 . . . . 5  |-  F/_ x NN
25 nfcv 2331 . . . . . . . 8  |-  F/_ x
f
26 nfcv 2331 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
2725, 26, 3nff1o 5473 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
28 nfcv 2331 . . . . . . . . . 10  |-  F/_ x
1
29 nfv 1538 . . . . . . . . . . . 12  |-  F/ x  n  <_  m
30 nfcv 2331 . . . . . . . . . . . . 13  |-  F/_ x
( f `  n
)
3130, 13nfcsb 3108 . . . . . . . . . . . 12  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
3229, 31, 15nfif 3576 . . . . . . . . . . 11  |-  F/_ x if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 )
3324, 32nfmpt 4109 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )
3428, 10, 33nfseq 10472 . . . . . . . . 9  |-  F/_ x  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) )
3534, 9nffv 5539 . . . . . . . 8  |-  F/_ x
(  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ) `  m
)
3635nfeq2 2343 . . . . . . 7  |-  F/ x  z  =  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ) `  m
)
3727, 36nfan 1575 . . . . . 6  |-  F/ x
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ) `  m
) )
3837nfex 1647 . . . . 5  |-  F/ x E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  z  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ) `  m
) )
3924, 38nfrexxy 2528 . . . 4  |-  F/ x E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  z  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ) `  m
) )
4023, 39nfor 1584 . . 3  |-  F/ x
( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  A. j  e.  (
ZZ>= `  m )DECID  j  e.  A  /\  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  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ) `  m
) ) )
4140nfiotaw 5196 . 2  |-  F/_ x
( iota z ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  A. j  e.  ( ZZ>= `  m )DECID  j  e.  A  /\  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  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ) `  m
) ) ) )
421, 41nfcxfr 2328 1  |-  F/_ x sum_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 104    \/ wo 709  DECID wdc 835    /\ w3a 979    = wceq 1363   E.wex 1502    e. wcel 2159   F/_wnfc 2318   A.wral 2467   E.wrex 2468   [_csb 3071    C_ wss 3143   ifcif 3548   class class class wbr 4017    |-> cmpt 4078   iotacio 5190   -1-1-onto->wf1o 5229   ` cfv 5230  (class class class)co 5890   0cc0 7828   1c1 7829    + caddc 7831    <_ cle 8010   NNcn 8936   ZZcz 9270   ZZ>=cuz 9545   ...cfz 10025    seqcseq 10462    ~~> cli 11303   sum_csu 11378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-un 3147  df-in 3149  df-ss 3156  df-if 3549  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-mpt 4080  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-ov 5893  df-oprab 5894  df-mpo 5895  df-recs 6323  df-frec 6409  df-seqfrec 10463  df-sumdc 11379
This theorem is referenced by:  fsum2dlemstep  11459  fisumcom2  11463  fsumiun  11502  fsumcncntop  14439
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