ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfsum Unicode version

Theorem nfsum 11133
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 11130 . 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 2281 . . . . 5  |-  F/_ x ZZ
3 nfsum.1 . . . . . . 7  |-  F/_ x A
4 nfcv 2281 . . . . . . 7  |-  F/_ x
( ZZ>= `  m )
53, 4nfss 3090 . . . . . 6  |-  F/ x  A  C_  ( ZZ>= `  m
)
63nfcri 2275 . . . . . . . 8  |-  F/ x  j  e.  A
76nfdc 1637 . . . . . . 7  |-  F/ xDECID  j  e.  A
84, 7nfralxy 2471 . . . . . 6  |-  F/ x A. j  e.  ( ZZ>=
`  m )DECID  j  e.  A
9 nfcv 2281 . . . . . . . 8  |-  F/_ x m
10 nfcv 2281 . . . . . . . 8  |-  F/_ x  +
113nfcri 2275 . . . . . . . . . 10  |-  F/ x  n  e.  A
12 nfcv 2281 . . . . . . . . . . 11  |-  F/_ x n
13 nfsum.2 . . . . . . . . . . 11  |-  F/_ x B
1412, 13nfcsb 3037 . . . . . . . . . 10  |-  F/_ x [_ n  /  k ]_ B
15 nfcv 2281 . . . . . . . . . 10  |-  F/_ x
0
1611, 14, 15nfif 3500 . . . . . . . . 9  |-  F/_ x if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
172, 16nfmpt 4020 . . . . . . . 8  |-  F/_ x
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) )
189, 10, 17nfseq 10235 . . . . . . 7  |-  F/_ x  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )
19 nfcv 2281 . . . . . . 7  |-  F/_ x  ~~>
20 nfcv 2281 . . . . . . 7  |-  F/_ x
z
2118, 19, 20nfbr 3974 . . . . . 6  |-  F/ x  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  z
225, 8, 21nf3an 1545 . . . . 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 2472 . . . 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 2281 . . . . 5  |-  F/_ x NN
25 nfcv 2281 . . . . . . . 8  |-  F/_ x
f
26 nfcv 2281 . . . . . . . 8  |-  F/_ x
( 1 ... m
)
2725, 26, 3nff1o 5365 . . . . . . 7  |-  F/ x  f : ( 1 ... m ) -1-1-onto-> A
28 nfcv 2281 . . . . . . . . . 10  |-  F/_ x
1
29 nfv 1508 . . . . . . . . . . . 12  |-  F/ x  n  <_  m
30 nfcv 2281 . . . . . . . . . . . . 13  |-  F/_ x
( f `  n
)
3130, 13nfcsb 3037 . . . . . . . . . . . 12  |-  F/_ x [_ ( f `  n
)  /  k ]_ B
3229, 31, 15nfif 3500 . . . . . . . . . . 11  |-  F/_ x if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 )
3324, 32nfmpt 4020 . . . . . . . . . 10  |-  F/_ x
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) )
3428, 10, 33nfseq 10235 . . . . . . . . 9  |-  F/_ x  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) )
3534, 9nffv 5431 . . . . . . . 8  |-  F/_ x
(  seq 1 (  +  ,  ( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n
)  /  k ]_ B ,  0 ) ) ) `  m
)
3635nfeq2 2293 . . . . . . 7  |-  F/ x  z  =  (  seq 1 (  +  , 
( n  e.  NN  |->  if ( n  <_  m ,  [_ ( f `  n )  /  k ]_ B ,  0 ) ) ) `  m
)
3727, 36nfan 1544 . . . . . 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 1616 . . . . 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 2472 . . . 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 1553 . . 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 5092 . 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 2278 1  |-  F/_ x sum_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 697  DECID wdc 819    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480   F/_wnfc 2268   A.wral 2416   E.wrex 2417   [_csb 3003    C_ wss 3071   ifcif 3474   class class class wbr 3929    |-> cmpt 3989   iotacio 5086   -1-1-onto->wf1o 5122   ` cfv 5123  (class class class)co 5774   0cc0 7627   1c1 7628    + caddc 7630    <_ cle 7808   NNcn 8727   ZZcz 9061   ZZ>=cuz 9333   ...cfz 9797    seqcseq 10225    ~~> cli 11054   sum_csu 11129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-seqfrec 10226  df-sumdc 11130
This theorem is referenced by:  fsum2dlemstep  11210  fisumcom2  11214  fsumiun  11253  fsumcncntop  12735
  Copyright terms: Public domain W3C validator