Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  esumfzf Structured version   Unicode version

Theorem esumfzf 28217
Description: Formulating a partial extended sum over integers using the recursive sequence builder. (Contributed by Thierry Arnoux, 18-Oct-2017.)
Hypothesis
Ref Expression
esumfzf.1  |-  F/_ k F
Assertion
Ref Expression
esumfzf  |-  ( ( F : NN --> ( 0 [,] +oo )  /\  N  e.  NN )  -> Σ* k  e.  ( 1 ... N ) ( F `
 k )  =  (  seq 1 ( +e ,  F
) `  N )
)
Distinct variable group:    k, N
Allowed substitution hint:    F( k)

Proof of Theorem esumfzf
Dummy variables  i  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1715 . . . . . 6  |-  F/ k  i  =  1
2 oveq2 6204 . . . . . 6  |-  ( i  =  1  ->  (
1 ... i )  =  ( 1 ... 1
) )
31, 2esumeq1d 28183 . . . . 5  |-  ( i  =  1  -> Σ* k  e.  ( 1 ... i ) ( F `  k
)  = Σ* k  e.  ( 1 ... 1 ) ( F `  k
) )
4 fveq2 5774 . . . . 5  |-  ( i  =  1  ->  (  seq 1 ( +e ,  F ) `  i
)  =  (  seq 1 ( +e ,  F ) `  1
) )
53, 4eqeq12d 2404 . . . 4  |-  ( i  =  1  ->  (Σ* k  e.  ( 1 ... i
) ( F `  k )  =  (  seq 1 ( +e ,  F ) `
 i )  <-> Σ* k  e.  ( 1 ... 1 ) ( F `  k
)  =  (  seq 1 ( +e ,  F ) `  1
) ) )
65imbi2d 314 . . 3  |-  ( i  =  1  ->  (
( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... i ) ( F `  k )  =  (  seq 1
( +e ,  F ) `  i
) )  <->  ( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... 1 ) ( F `  k
)  =  (  seq 1 ( +e ,  F ) `  1
) ) ) )
7 nfv 1715 . . . . . 6  |-  F/ k  i  =  n
8 oveq2 6204 . . . . . 6  |-  ( i  =  n  ->  (
1 ... i )  =  ( 1 ... n
) )
97, 8esumeq1d 28183 . . . . 5  |-  ( i  =  n  -> Σ* k  e.  ( 1 ... i ) ( F `  k
)  = Σ* k  e.  ( 1 ... n ) ( F `  k
) )
10 fveq2 5774 . . . . 5  |-  ( i  =  n  ->  (  seq 1 ( +e ,  F ) `  i
)  =  (  seq 1 ( +e ,  F ) `  n
) )
119, 10eqeq12d 2404 . . . 4  |-  ( i  =  n  ->  (Σ* k  e.  ( 1 ... i
) ( F `  k )  =  (  seq 1 ( +e ,  F ) `
 i )  <-> Σ* k  e.  ( 1 ... n ) ( F `  k
)  =  (  seq 1 ( +e ,  F ) `  n
) ) )
1211imbi2d 314 . . 3  |-  ( i  =  n  ->  (
( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... i ) ( F `  k )  =  (  seq 1
( +e ,  F ) `  i
) )  <->  ( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... n ) ( F `  k
)  =  (  seq 1 ( +e ,  F ) `  n
) ) ) )
13 nfv 1715 . . . . . 6  |-  F/ k  i  =  ( n  +  1 )
14 oveq2 6204 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  (
1 ... i )  =  ( 1 ... (
n  +  1 ) ) )
1513, 14esumeq1d 28183 . . . . 5  |-  ( i  =  ( n  + 
1 )  -> Σ* k  e.  ( 1 ... i ) ( F `  k
)  = Σ* k  e.  ( 1 ... ( n  +  1 ) ) ( F `  k
) )
16 fveq2 5774 . . . . 5  |-  ( i  =  ( n  + 
1 )  ->  (  seq 1 ( +e ,  F ) `  i
)  =  (  seq 1 ( +e ,  F ) `  (
n  +  1 ) ) )
1715, 16eqeq12d 2404 . . . 4  |-  ( i  =  ( n  + 
1 )  ->  (Σ* k  e.  ( 1 ... i
) ( F `  k )  =  (  seq 1 ( +e ,  F ) `
 i )  <-> Σ* k  e.  ( 1 ... ( n  +  1 ) ) ( F `  k
)  =  (  seq 1 ( +e ,  F ) `  (
n  +  1 ) ) ) )
1817imbi2d 314 . . 3  |-  ( i  =  ( n  + 
1 )  ->  (
( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... i ) ( F `  k )  =  (  seq 1
( +e ,  F ) `  i
) )  <->  ( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... ( n  +  1 ) ) ( F `  k
)  =  (  seq 1 ( +e ,  F ) `  (
n  +  1 ) ) ) ) )
19 nfv 1715 . . . . . 6  |-  F/ k  i  =  N
20 oveq2 6204 . . . . . 6  |-  ( i  =  N  ->  (
1 ... i )  =  ( 1 ... N
) )
2119, 20esumeq1d 28183 . . . . 5  |-  ( i  =  N  -> Σ* k  e.  ( 1 ... i ) ( F `  k
)  = Σ* k  e.  ( 1 ... N ) ( F `  k
) )
22 fveq2 5774 . . . . 5  |-  ( i  =  N  ->  (  seq 1 ( +e ,  F ) `  i
)  =  (  seq 1 ( +e ,  F ) `  N
) )
2321, 22eqeq12d 2404 . . . 4  |-  ( i  =  N  ->  (Σ* k  e.  ( 1 ... i
) ( F `  k )  =  (  seq 1 ( +e ,  F ) `
 i )  <-> Σ* k  e.  ( 1 ... N ) ( F `  k
)  =  (  seq 1 ( +e ,  F ) `  N
) ) )
2423imbi2d 314 . . 3  |-  ( i  =  N  ->  (
( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... i ) ( F `  k )  =  (  seq 1
( +e ,  F ) `  i
) )  <->  ( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... N ) ( F `  k
)  =  (  seq 1 ( +e ,  F ) `  N
) ) ) )
25 fveq2 5774 . . . . . 6  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
26 nfcv 2544 . . . . . 6  |-  F/_ x { 1 }
27 nfcv 2544 . . . . . 6  |-  F/_ k { 1 }
28 nfcv 2544 . . . . . 6  |-  F/_ x
( F `  k
)
29 esumfzf.1 . . . . . . 7  |-  F/_ k F
30 nfcv 2544 . . . . . . 7  |-  F/_ k
x
3129, 30nffv 5781 . . . . . 6  |-  F/_ k
( F `  x
)
3225, 26, 27, 28, 31cbvesum 28190 . . . . 5  |- Σ* k  e.  {
1 }  ( F `
 k )  = Σ* x  e.  { 1 }  ( F `  x
)
33 simpr 459 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,] +oo )  /\  x  =  1 )  ->  x  =  1 )
3433fveq2d 5778 . . . . . 6  |-  ( ( F : NN --> ( 0 [,] +oo )  /\  x  =  1 )  ->  ( F `  x )  =  ( F `  1 ) )
35 1z 10811 . . . . . . 7  |-  1  e.  ZZ
3635a1i 11 . . . . . 6  |-  ( F : NN --> ( 0 [,] +oo )  -> 
1  e.  ZZ )
37 1nn 10463 . . . . . . 7  |-  1  e.  NN
38 ffvelrn 5931 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,] +oo )  /\  1  e.  NN )  ->  ( F `  1
)  e.  ( 0 [,] +oo ) )
3937, 38mpan2 669 . . . . . 6  |-  ( F : NN --> ( 0 [,] +oo )  -> 
( F `  1
)  e.  ( 0 [,] +oo ) )
4034, 36, 39esumsn 28213 . . . . 5  |-  ( F : NN --> ( 0 [,] +oo )  -> Σ* x  e.  { 1 }  ( F `  x )  =  ( F ` 
1 ) )
4132, 40syl5eq 2435 . . . 4  |-  ( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  { 1 }  ( F `  k )  =  ( F ` 
1 ) )
42 fzsn 11647 . . . . . 6  |-  ( 1  e.  ZZ  ->  (
1 ... 1 )  =  { 1 } )
4335, 42ax-mp 5 . . . . 5  |-  ( 1 ... 1 )  =  { 1 }
44 esumeq1 28182 . . . . 5  |-  ( ( 1 ... 1 )  =  { 1 }  -> Σ* k  e.  ( 1 ... 1 ) ( F `  k )  = Σ* k  e.  { 1 }  ( F `  k ) )
4543, 44ax-mp 5 . . . 4  |- Σ* k  e.  ( 1 ... 1 ) ( F `  k
)  = Σ* k  e.  {
1 }  ( F `
 k )
46 seq1 12023 . . . . 5  |-  ( 1  e.  ZZ  ->  (  seq 1 ( +e ,  F ) `  1
)  =  ( F `
 1 ) )
4735, 46ax-mp 5 . . . 4  |-  (  seq 1 ( +e ,  F ) `  1
)  =  ( F `
 1 )
4841, 45, 473eqtr4g 2448 . . 3  |-  ( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... 1
) ( F `  k )  =  (  seq 1 ( +e ,  F ) `
 1 ) )
49 simpl 455 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  n  e.  NN )
50 nnuz 11036 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
5149, 50syl6eleq 2480 . . . . . . . 8  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  n  e.  (
ZZ>= `  1 ) )
52 seqp1 12025 . . . . . . . 8  |-  ( n  e.  ( ZZ>= `  1
)  ->  (  seq 1 ( +e ,  F ) `  (
n  +  1 ) )  =  ( (  seq 1 ( +e ,  F ) `
 n ) +e ( F `  ( n  +  1
) ) ) )
5351, 52syl 16 . . . . . . 7  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  (  seq 1
( +e ,  F ) `  (
n  +  1 ) )  =  ( (  seq 1 ( +e ,  F ) `
 n ) +e ( F `  ( n  +  1
) ) ) )
5453adantr 463 . . . . . 6  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\ Σ* k  e.  ( 1 ... n ) ( F `  k )  =  (  seq 1
( +e ,  F ) `  n
) )  ->  (  seq 1 ( +e ,  F ) `  (
n  +  1 ) )  =  ( (  seq 1 ( +e ,  F ) `
 n ) +e ( F `  ( n  +  1
) ) ) )
55 simpr 459 . . . . . . 7  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\ Σ* k  e.  ( 1 ... n ) ( F `  k )  =  (  seq 1
( +e ,  F ) `  n
) )  -> Σ* k  e.  ( 1 ... n ) ( F `  k
)  =  (  seq 1 ( +e ,  F ) `  n
) )
5655oveq1d 6211 . . . . . 6  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\ Σ* k  e.  ( 1 ... n ) ( F `  k )  =  (  seq 1
( +e ,  F ) `  n
) )  ->  (Σ* k  e.  ( 1 ... n
) ( F `  k ) +e
( F `  (
n  +  1 ) ) )  =  ( (  seq 1 ( +e ,  F
) `  n ) +e ( F `
 ( n  + 
1 ) ) ) )
57 nfv 1715 . . . . . . . . . 10  |-  F/ k  n  e.  NN
5857nfci 2533 . . . . . . . . . . 11  |-  F/_ k NN
59 nfcv 2544 . . . . . . . . . . 11  |-  F/_ k
( 0 [,] +oo )
6029, 58, 59nff 5635 . . . . . . . . . 10  |-  F/ k  F : NN --> ( 0 [,] +oo )
6157, 60nfan 1936 . . . . . . . . 9  |-  F/ k ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )
62 fzsuc 11649 . . . . . . . . . 10  |-  ( n  e.  ( ZZ>= `  1
)  ->  ( 1 ... ( n  + 
1 ) )  =  ( ( 1 ... n )  u.  {
( n  +  1 ) } ) )
6351, 62syl 16 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  ( 1 ... ( n  +  1 ) )  =  ( ( 1 ... n
)  u.  { ( n  +  1 ) } ) )
6461, 63esumeq1d 28183 . . . . . . . 8  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  -> Σ* k  e.  ( 1 ... ( n  + 
1 ) ) ( F `  k )  = Σ* k  e.  ( ( 1 ... n )  u.  { ( n  +  1 ) } ) ( F `  k ) )
65 nfcv 2544 . . . . . . . . 9  |-  F/_ k
( 1 ... n
)
66 nfcv 2544 . . . . . . . . 9  |-  F/_ k { ( n  + 
1 ) }
67 ovex 6224 . . . . . . . . . 10  |-  ( 1 ... n )  e. 
_V
6867a1i 11 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  ( 1 ... n )  e.  _V )
69 snex 4603 . . . . . . . . . 10  |-  { ( n  +  1 ) }  e.  _V
7069a1i 11 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  { ( n  +  1 ) }  e.  _V )
71 fzp1disj 11660 . . . . . . . . . 10  |-  ( ( 1 ... n )  i^i  { ( n  +  1 ) } )  =  (/)
7271a1i 11 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  ( ( 1 ... n )  i^i 
{ ( n  + 
1 ) } )  =  (/) )
73 simplr 753 . . . . . . . . . 10  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  ( 1 ... n ) )  ->  F : NN
--> ( 0 [,] +oo ) )
74 fzssnn 27748 . . . . . . . . . . . 12  |-  ( 1  e.  NN  ->  (
1 ... n )  C_  NN )
7537, 74ax-mp 5 . . . . . . . . . . 11  |-  ( 1 ... n )  C_  NN
76 simpr 459 . . . . . . . . . . 11  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  ( 1 ... n ) )  ->  k  e.  ( 1 ... n
) )
7775, 76sseldi 3415 . . . . . . . . . 10  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  ( 1 ... n ) )  ->  k  e.  NN )
7873, 77ffvelrnd 5934 . . . . . . . . 9  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  ( 1 ... n ) )  ->  ( F `  k )  e.  ( 0 [,] +oo )
)
79 simplr 753 . . . . . . . . . 10  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  {
( n  +  1 ) } )  ->  F : NN --> ( 0 [,] +oo ) )
80 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  {
( n  +  1 ) } )  -> 
k  e.  { ( n  +  1 ) } )
81 elsn 3958 . . . . . . . . . . . 12  |-  ( k  e.  { ( n  +  1 ) }  <-> 
k  =  ( n  +  1 ) )
8280, 81sylib 196 . . . . . . . . . . 11  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  {
( n  +  1 ) } )  -> 
k  =  ( n  +  1 ) )
83 simpll 751 . . . . . . . . . . . 12  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  {
( n  +  1 ) } )  ->  n  e.  NN )
8483peano2nnd 10469 . . . . . . . . . . 11  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  {
( n  +  1 ) } )  -> 
( n  +  1 )  e.  NN )
8582, 84eqeltrd 2470 . . . . . . . . . 10  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  {
( n  +  1 ) } )  -> 
k  e.  NN )
8679, 85ffvelrnd 5934 . . . . . . . . 9  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  {
( n  +  1 ) } )  -> 
( F `  k
)  e.  ( 0 [,] +oo ) )
8761, 65, 66, 68, 70, 72, 78, 86esumsplit 28201 . . . . . . . 8  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  -> Σ* k  e.  ( ( 1 ... n )  u.  { ( n  +  1 ) } ) ( F `  k )  =  (Σ* k  e.  ( 1 ... n ) ( F `
 k ) +eΣ* k  e.  { ( n  +  1 ) }  ( F `  k ) ) )
88 nfcv 2544 . . . . . . . . . . 11  |-  F/_ x { ( n  + 
1 ) }
8925, 88, 66, 28, 31cbvesum 28190 . . . . . . . . . 10  |- Σ* k  e.  {
( n  +  1 ) }  ( F `
 k )  = Σ* x  e.  { ( n  +  1 ) }  ( F `  x
)
90 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  x  =  ( n  +  1 ) )  ->  x  =  ( n  +  1
) )
9190fveq2d 5778 . . . . . . . . . . 11  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  x  =  ( n  +  1 ) )  ->  ( F `  x )  =  ( F `  ( n  +  1 ) ) )
9249peano2nnd 10469 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  ( n  + 
1 )  e.  NN )
93 simpr 459 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  F : NN --> ( 0 [,] +oo ) )
9493, 92ffvelrnd 5934 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  ( F `  ( n  +  1
) )  e.  ( 0 [,] +oo )
)
9591, 92, 94esumsn 28213 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  -> Σ* x  e.  { ( n  +  1 ) }  ( F `  x )  =  ( F `  ( n  +  1 ) ) )
9689, 95syl5eq 2435 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  -> Σ* k  e.  { ( n  +  1 ) }  ( F `  k )  =  ( F `  ( n  +  1 ) ) )
9796oveq2d 6212 . . . . . . . 8  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  (Σ* k  e.  ( 1 ... n ) ( F `  k ) +eΣ* k  e.  { ( n  +  1 ) }  ( F `  k ) )  =  (Σ* k  e.  ( 1 ... n ) ( F `  k ) +e ( F `
 ( n  + 
1 ) ) ) )
9864, 87, 973eqtrrd 2428 . . . . . . 7  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  (Σ* k  e.  ( 1 ... n ) ( F `  k ) +e ( F `
 ( n  + 
1 ) ) )  = Σ* k  e.  ( 1 ... ( n  + 
1 ) ) ( F `  k ) )
9998adantr 463 . . . . . 6  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\ Σ* k  e.  ( 1 ... n ) ( F `  k )  =  (  seq 1
( +e ,  F ) `  n
) )  ->  (Σ* k  e.  ( 1 ... n
) ( F `  k ) +e
( F `  (
n  +  1 ) ) )  = Σ* k  e.  ( 1 ... (
n  +  1 ) ) ( F `  k ) )
10054, 56, 993eqtr2rd 2430 . . . . 5  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\ Σ* k  e.  ( 1 ... n ) ( F `  k )  =  (  seq 1
( +e ,  F ) `  n
) )  -> Σ* k  e.  ( 1 ... ( n  +  1 ) ) ( F `  k
)  =  (  seq 1 ( +e ,  F ) `  (
n  +  1 ) ) )
101100exp31 602 . . . 4  |-  ( n  e.  NN  ->  ( F : NN --> ( 0 [,] +oo )  -> 
(Σ* k  e.  ( 1 ... n ) ( F `  k )  =  (  seq 1
( +e ,  F ) `  n
)  -> Σ* k  e.  ( 1 ... ( n  +  1 ) ) ( F `  k
)  =  (  seq 1 ( +e ,  F ) `  (
n  +  1 ) ) ) ) )
102101a2d 26 . . 3  |-  ( n  e.  NN  ->  (
( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... n ) ( F `  k )  =  (  seq 1
( +e ,  F ) `  n
) )  ->  ( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... (
n  +  1 ) ) ( F `  k )  =  (  seq 1 ( +e ,  F ) `
 ( n  + 
1 ) ) ) ) )
1036, 12, 18, 24, 48, 102nnind 10470 . 2  |-  ( N  e.  NN  ->  ( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... N
) ( F `  k )  =  (  seq 1 ( +e ,  F ) `
 N ) ) )
104103impcom 428 1  |-  ( ( F : NN --> ( 0 [,] +oo )  /\  N  e.  NN )  -> Σ* k  e.  ( 1 ... N ) ( F `
 k )  =  (  seq 1 ( +e ,  F
) `  N )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   F/_wnfc 2530   _Vcvv 3034    u. cun 3387    i^i cin 3388    C_ wss 3389   (/)c0 3711   {csn 3944   -->wf 5492   ` cfv 5496  (class class class)co 6196   0cc0 9403   1c1 9404    + caddc 9406   +oocpnf 9536   NNcn 10452   ZZcz 10781   ZZ>=cuz 11001   +ecxad 11237   [,]cicc 11453   ...cfz 11593    seqcseq 12010  Σ*cesum 28175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ioc 11455  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070  df-fac 12256  df-bc 12283  df-hash 12308  df-shft 12902  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-limsup 13296  df-clim 13313  df-rlim 13314  df-sum 13511  df-ef 13805  df-sin 13807  df-cos 13808  df-pi 13810  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-rest 14830  df-topn 14831  df-0g 14849  df-gsum 14850  df-topgen 14851  df-pt 14852  df-prds 14855  df-ordt 14908  df-xrs 14909  df-qtop 14914  df-imas 14915  df-xps 14917  df-mre 14993  df-mrc 14994  df-acs 14996  df-ps 15947  df-tsr 15948  df-plusf 15988  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mhm 16083  df-submnd 16084  df-grp 16174  df-minusg 16175  df-sbg 16176  df-mulg 16177  df-subg 16315  df-cntz 16472  df-cmn 16917  df-abl 16918  df-mgp 17255  df-ur 17267  df-ring 17313  df-cring 17314  df-subrg 17540  df-abv 17579  df-lmod 17627  df-scaf 17628  df-sra 17931  df-rgmod 17932  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-fbas 18529  df-fg 18530  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cld 19605  df-ntr 19606  df-cls 19607  df-nei 19685  df-lp 19723  df-perf 19724  df-cn 19814  df-cnp 19815  df-haus 19902  df-tx 20148  df-hmeo 20341  df-fil 20432  df-fm 20524  df-flim 20525  df-flf 20526  df-tmd 20656  df-tgp 20657  df-tsms 20710  df-trg 20747  df-xms 20908  df-ms 20909  df-tms 20910  df-nm 21188  df-ngp 21189  df-nrg 21191  df-nlm 21192  df-ii 21466  df-cncf 21467  df-limc 22355  df-dv 22356  df-log 23029  df-esum 28176
This theorem is referenced by:  esumfsup  28218  esumsup  28237
  Copyright terms: Public domain W3C validator