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

Theorem nfsum1 12447
Description: Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
Hypothesis
Ref Expression
nfsum1.1  |-  F/_ k A
Assertion
Ref Expression
nfsum1  |-  F/_ k sum_ k  e.  A  B
Distinct variable group:    A, k
Allowed substitution hint:    B( k)

Proof of Theorem nfsum1
Dummy variables  f  m  n  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sum 12443 . 2  |-  sum_ k  e.  A  B  =  ( iota y ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  y )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
2 nfcv 2548 . . . . 5  |-  F/_ k ZZ
3 nfsum1.1 . . . . . . 7  |-  F/_ k A
4 nfcv 2548 . . . . . . 7  |-  F/_ k
( ZZ>= `  m )
53, 4nfss 3309 . . . . . 6  |-  F/ k  A  C_  ( ZZ>= `  m )
6 nfcv 2548 . . . . . . . 8  |-  F/_ k
m
7 nfcv 2548 . . . . . . . 8  |-  F/_ k  +
8 nfmpt1 4266 . . . . . . . 8  |-  F/_ k
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
96, 7, 8nfseq 11296 . . . . . . 7  |-  F/_ k  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )
10 nfcv 2548 . . . . . . 7  |-  F/_ k  ~~>
11 nfcv 2548 . . . . . . 7  |-  F/_ k
y
129, 10, 11nfbr 4224 . . . . . 6  |-  F/ k  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  y
135, 12nfan 1842 . . . . 5  |-  F/ k ( A  C_  ( ZZ>=
`  m )  /\  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  y )
142, 13nfrex 2729 . . . 4  |-  F/ k E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  y )
15 nfcv 2548 . . . . 5  |-  F/_ k NN
16 nfcv 2548 . . . . . . . 8  |-  F/_ k
f
17 nfcv 2548 . . . . . . . 8  |-  F/_ k
( 1 ... m
)
1816, 17, 3nff1o 5639 . . . . . . 7  |-  F/ k  f : ( 1 ... m ) -1-1-onto-> A
19 nfcv 2548 . . . . . . . . . 10  |-  F/_ k
1
20 nfcsb1v 3251 . . . . . . . . . . 11  |-  F/_ k [_ ( f `  n
)  /  k ]_ B
2115, 20nfmpt 4265 . . . . . . . . . 10  |-  F/_ k
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
2219, 7, 21nfseq 11296 . . . . . . . . 9  |-  F/_ k  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
2322, 6nffv 5702 . . . . . . . 8  |-  F/_ k
(  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
)
2423nfeq2 2559 . . . . . . 7  |-  F/ k  y  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
2518, 24nfan 1842 . . . . . 6  |-  F/ k ( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
2625nfex 1861 . . . . 5  |-  F/ k E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
) )
2715, 26nfrex 2729 . . . 4  |-  F/ k E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
2814, 27nfor 1854 . . 3  |-  F/ k ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  y )  \/ 
E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) ) )
2928nfiota 5389 . 2  |-  F/_ k
( iota y ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  y )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
301, 29nfcxfr 2545 1  |-  F/_ k sum_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    \/ wo 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   F/_wnfc 2535   E.wrex 2675   [_csb 3219    C_ wss 3288   ifcif 3707   class class class wbr 4180    e. cmpt 4234   iotacio 5383   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6048   0cc0 8954   1c1 8955    + caddc 8957   NNcn 9964   ZZcz 10246   ZZ>=cuz 10452   ...cfz 11007    seq cseq 11286    ~~> cli 12241   sum_csu 12442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-recs 6600  df-rdg 6635  df-seq 11287  df-sum 12443
  Copyright terms: Public domain W3C validator