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Theorem nfsum1 12515
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 12511 . 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 2578 . . . . 5  |-  F/_ k ZZ
3 nfsum1.1 . . . . . . 7  |-  F/_ k A
4 nfcv 2578 . . . . . . 7  |-  F/_ k
( ZZ>= `  m )
53, 4nfss 3327 . . . . . 6  |-  F/ k  A  C_  ( ZZ>= `  m )
6 nfcv 2578 . . . . . . . 8  |-  F/_ k
m
7 nfcv 2578 . . . . . . . 8  |-  F/_ k  +
8 nfmpt1 4323 . . . . . . . 8  |-  F/_ k
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
96, 7, 8nfseq 11364 . . . . . . 7  |-  F/_ k  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )
10 nfcv 2578 . . . . . . 7  |-  F/_ k  ~~>
11 nfcv 2578 . . . . . . 7  |-  F/_ k
y
129, 10, 11nfbr 4281 . . . . . 6  |-  F/ k  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  y
135, 12nfan 1848 . . . . 5  |-  F/ k ( A  C_  ( ZZ>=
`  m )  /\  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  y )
142, 13nfrex 2767 . . . 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 2578 . . . . 5  |-  F/_ k NN
16 nfcv 2578 . . . . . . . 8  |-  F/_ k
f
17 nfcv 2578 . . . . . . . 8  |-  F/_ k
( 1 ... m
)
1816, 17, 3nff1o 5701 . . . . . . 7  |-  F/ k  f : ( 1 ... m ) -1-1-onto-> A
19 nfcv 2578 . . . . . . . . . 10  |-  F/_ k
1
20 nfcsb1v 3282 . . . . . . . . . . 11  |-  F/_ k [_ ( f `  n
)  /  k ]_ B
2115, 20nfmpt 4322 . . . . . . . . . 10  |-  F/_ k
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
2219, 7, 21nfseq 11364 . . . . . . . . 9  |-  F/_ k  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
2322, 6nffv 5764 . . . . . . . 8  |-  F/_ k
(  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
)
2423nfeq2 2589 . . . . . . 7  |-  F/ k  y  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
2518, 24nfan 1848 . . . . . 6  |-  F/ k ( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
2625nfex 1867 . . . . 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 2767 . . . 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 1860 . . 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 5451 . 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 2575 1  |-  F/_ k sum_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    \/ wo 359    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1727   F/_wnfc 2565   E.wrex 2712   [_csb 3267    C_ wss 3306   ifcif 3763   class class class wbr 4237    e. cmpt 4291   iotacio 5445   -1-1-onto->wf1o 5482   ` cfv 5483  (class class class)co 6110   0cc0 9021   1c1 9022    + caddc 9024   NNcn 10031   ZZcz 10313   ZZ>=cuz 10519   ...cfz 11074    seq cseq 11354    ~~> cli 12309   sum_csu 12510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-recs 6662  df-rdg 6697  df-seq 11355  df-sum 12511
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