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Theorem nfsum1 12254
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 12250 . 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 2494 . . . . 5  |-  F/_ k ZZ
3 nfsum1.1 . . . . . . 7  |-  F/_ k A
4 nfcv 2494 . . . . . . 7  |-  F/_ k
( ZZ>= `  m )
53, 4nfss 3249 . . . . . 6  |-  F/ k  A  C_  ( ZZ>= `  m )
6 nfcv 2494 . . . . . . . 8  |-  F/_ k
m
7 nfcv 2494 . . . . . . . 8  |-  F/_ k  +
8 nfmpt1 4188 . . . . . . . 8  |-  F/_ k
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
96, 7, 8nfseq 11145 . . . . . . 7  |-  F/_ k  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )
10 nfcv 2494 . . . . . . 7  |-  F/_ k  ~~>
11 nfcv 2494 . . . . . . 7  |-  F/_ k
y
129, 10, 11nfbr 4146 . . . . . 6  |-  F/ k  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  y
135, 12nfan 1829 . . . . 5  |-  F/ k ( A  C_  ( ZZ>=
`  m )  /\  seq  m (  +  , 
( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  y )
142, 13nfrex 2674 . . . 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 2494 . . . . 5  |-  F/_ k NN
16 nfcv 2494 . . . . . . . 8  |-  F/_ k
f
17 nfcv 2494 . . . . . . . 8  |-  F/_ k
( 1 ... m
)
1816, 17, 3nff1o 5550 . . . . . . 7  |-  F/ k  f : ( 1 ... m ) -1-1-onto-> A
19 nfcv 2494 . . . . . . . . . 10  |-  F/_ k
1
20 nfcsb1v 3189 . . . . . . . . . . 11  |-  F/_ k [_ ( f `  n
)  /  k ]_ B
2115, 20nfmpt 4187 . . . . . . . . . 10  |-  F/_ k
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
2219, 7, 21nfseq 11145 . . . . . . . . 9  |-  F/_ k  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
2322, 6nffv 5612 . . . . . . . 8  |-  F/_ k
(  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
)
2423nfeq2 2505 . . . . . . 7  |-  F/ k  y  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
2518, 24nfan 1829 . . . . . 6  |-  F/ k ( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
2625nfex 1848 . . . . 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 2674 . . . 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 1841 . . 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 5302 . 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 2491 1  |-  F/_ k sum_ k  e.  A  B
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358   E.wex 1541    = wceq 1642    e. wcel 1710   F/_wnfc 2481   E.wrex 2620   [_csb 3157    C_ wss 3228   ifcif 3641   class class class wbr 4102    e. cmpt 4156   iotacio 5296   -1-1-onto->wf1o 5333   ` cfv 5334  (class class class)co 5942   0cc0 8824   1c1 8825    + caddc 8827   NNcn 9833   ZZcz 10113   ZZ>=cuz 10319   ...cfz 10871    seq cseq 11135    ~~> cli 12048   sum_csu 12249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-recs 6472  df-rdg 6507  df-seq 11136  df-sum 12250
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