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Theorem nfsum 10332
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-isum 10329 . 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 ) ) ,  CC )  ~~>  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 ) ) ,  CC ) `
 m ) ) ) )
2 nfcv 2220 . . . . 5  |-  F/_ x ZZ
3 nfsum.1 . . . . . . 7  |-  F/_ x A
4 nfcv 2220 . . . . . . 7  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 2993 . . . . . 6  |-  F/ x  A  C_  ( ZZ>= `  m
)
63nfcri 2214 . . . . . . . 8  |-  F/ x  j  e.  A
76nfdc 1590 . . . . . . 7  |-  F/ xDECID  j  e.  A
84, 7nfralxy 2403 . . . . . 6  |-  F/ x A. j  e.  ( ZZ>=
`  m )DECID  j  e.  A
9 nfcv 2220 . . . . . . . 8  |-  F/_ x m
10 nfcv 2220 . . . . . . . 8  |-  F/_ x  +
113nfcri 2214 . . . . . . . . . 10  |-  F/ x  n  e.  A
12 nfcv 2220 . . . . . . . . . . 11  |-  F/_ x n
13 nfsum.2 . . . . . . . . . . 11  |-  F/_ x B
1412, 13nfcsb 2941 . . . . . . . . . 10  |-  F/_ x [_ n  /  k ]_ B
15 nfcv 2220 . . . . . . . . . 10  |-  F/_ x
0
1611, 14, 15nfif 3385 . . . . . . . . 9  |-  F/_ x if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
172, 16nfmpt 3878 . . . . . . . 8  |-  F/_ x
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) )
18 nfcv 2220 . . . . . . . 8  |-  F/_ x CC
199, 10, 17, 18nfiseq 9528 . . . . . . 7  |-  F/_ x  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )
20 nfcv 2220 . . . . . . 7  |-  F/_ x  ~~>
21 nfcv 2220 . . . . . . 7  |-  F/_ x
z
2219, 20, 21nfbr 3837 . . . . . 6  |-  F/ x  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) ,  CC )  ~~>  z
235, 8, 22nf3an 1499 . . . . 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 ) ) ,  CC )  ~~>  z )
242, 23nfrexxy 2404 . . . 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 ) ) ,  CC )  ~~>  z )
25 nfcv 2220 . . . . 5  |-  F/_ x NN
26 nfcv 2220 . . . . . . . 8  |-  F/_ x
f
27 nfcv 2220 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
2826, 27, 3nff1o 5155 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
29 nfcv 2220 . . . . . . . . . 10  |-  F/_ x
1
30 nfv 1462 . . . . . . . . . . . 12  |-  F/ x  n  <_  m
31 nfcv 2220 . . . . . . . . . . . . 13  |-  F/_ x
( f `  n
)
3231, 13nfcsb 2941 . . . . . . . . . . . 12  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
3330, 32, 15nfif 3385 . . . . . . . . . . 11  |-  F/_ x if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 )
3425, 33nfmpt 3878 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )
3529, 10, 34, 18nfiseq 9528 . . . . . . . . 9  |-  F/_ x  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ,  CC )
3635, 9nffv 5216 . . . . . . . 8  |-  F/_ x
(  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m )
3736nfeq2 2231 . . . . . . 7  |-  F/ x  z  =  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ,  CC ) `
 m )
3828, 37nfan 1498 . . . . . 6  |-  F/ x
( f : ( 1 ... m ) -1-1-onto-> A  /\  z  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ,  CC ) `
 m ) )
3938nfex 1569 . . . . 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 ) ) ,  CC ) `
 m ) )
4025, 39nfrexxy 2404 . . . 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 ) ) ,  CC ) `
 m ) )
4124, 40nfor 1507 . . 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 ) ) ,  CC )  ~~>  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 ) ) ,  CC ) `
 m ) ) )
4241nfiotaxy 4901 . 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 ) ) ,  CC )  ~~>  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 ) ) ,  CC ) `
 m ) ) ) )
431, 42nfcxfr 2217 1  |-  F/_ x sum_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 102    \/ wo 662  DECID wdc 776    /\ w3a 920    = wceq 1285   E.wex 1422    e. wcel 1434   F/_wnfc 2207   A.wral 2349   E.wrex 2350   [_csb 2909    C_ wss 2974   ifcif 3359   class class class wbr 3793    |-> cmpt 3847   iotacio 4895   -1-1-onto->wf1o 4931   ` cfv 4932  (class class class)co 5543   CCcc 7041   0cc0 7043   1c1 7044    + caddc 7046    <_ cle 7216   NNcn 8106   ZZcz 8432   ZZ>=cuz 8700   ...cfz 9105    seqcseq 9521    ~~> cli 10255   sum_csu 10328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-if 3360  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-recs 5954  df-frec 6040  df-iseq 9522  df-isum 10329
This theorem is referenced by: (None)
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