MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfsum1 Structured version   Unicode version

Theorem nfsum1 13163
Description: Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Hypothesis
Ref Expression
nfsum1.1  |-  F/_ k A
Assertion
Ref Expression
nfsum1  |-  F/_ k sum_ k  e.  A  B

Proof of Theorem nfsum1
Dummy variables  f  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sum 13160 . 2  |-  sum_ k  e.  A  B  =  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
2 nfcv 2577 . . . . 5  |-  F/_ k ZZ
3 nfsum1.1 . . . . . . 7  |-  F/_ k A
4 nfcv 2577 . . . . . . 7  |-  F/_ k
( ZZ>= `  m )
53, 4nfss 3346 . . . . . 6  |-  F/ k  A  C_  ( ZZ>= `  m )
6 nfcv 2577 . . . . . . . 8  |-  F/_ k
m
7 nfcv 2577 . . . . . . . 8  |-  F/_ k  +
83nfel2 2589 . . . . . . . . . 10  |-  F/ k  n  e.  A
9 nfcsb1v 3301 . . . . . . . . . 10  |-  F/_ k [_ n  /  k ]_ B
10 nfcv 2577 . . . . . . . . . 10  |-  F/_ k
0
118, 9, 10nfif 3815 . . . . . . . . 9  |-  F/_ k if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 )
122, 11nfmpt 4377 . . . . . . . 8  |-  F/_ k
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) )
136, 7, 12nfseq 11812 . . . . . . 7  |-  F/_ k  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )
14 nfcv 2577 . . . . . . 7  |-  F/_ k  ~~>
15 nfcv 2577 . . . . . . 7  |-  F/_ k
x
1613, 14, 15nfbr 4333 . . . . . 6  |-  F/ k  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x
175, 16nfan 1865 . . . . 5  |-  F/ k ( A  C_  ( ZZ>=
`  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )
182, 17nfrex 2769 . . . 4  |-  F/ k E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  +  , 
( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )
19 nfcv 2577 . . . . 5  |-  F/_ k NN
20 nfcv 2577 . . . . . . . 8  |-  F/_ k
f
21 nfcv 2577 . . . . . . . 8  |-  F/_ k
( 1 ... m
)
2220, 21, 3nff1o 5636 . . . . . . 7  |-  F/ k  f : ( 1 ... m ) -1-1-onto-> A
23 nfcv 2577 . . . . . . . . . 10  |-  F/_ k
1
24 nfcsb1v 3301 . . . . . . . . . . 11  |-  F/_ k [_ ( f `  n
)  /  k ]_ B
2519, 24nfmpt 4377 . . . . . . . . . 10  |-  F/_ k
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
2623, 7, 25nfseq 11812 . . . . . . . . 9  |-  F/_ k  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
2726, 6nffv 5695 . . . . . . . 8  |-  F/_ k
(  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m )
2827nfeq2 2588 . . . . . . 7  |-  F/ k  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
2922, 28nfan 1865 . . . . . 6  |-  F/ k ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3029nfex 1878 . . . . 5  |-  F/ k E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
) )
3119, 30nfrex 2769 . . . 4  |-  F/ k E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
3218, 31nfor 1872 . . 3  |-  F/ k ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq m (  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )  \/ 
E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq 1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) ) )
3332nfiota 5382 . 2  |-  F/_ k
( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq m
(  +  ,  ( n  e.  ZZ  |->  if ( n  e.  A ,  [_ n  /  k ]_ B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq 1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
341, 33nfcxfr 2574 1  |-  F/_ k sum_ k  e.  A  B
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    = wceq 1364   E.wex 1591    e. wcel 1761   F/_wnfc 2564   E.wrex 2714   [_csb 3285    C_ wss 3325   ifcif 3788   class class class wbr 4289    e. cmpt 4347   iotacio 5376   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279    + caddc 9281   NNcn 10318   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433    seqcseq 11802    ~~> cli 12958   sum_csu 13159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-recs 6828  df-rdg 6862  df-seq 11803  df-sum 13160
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator