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Theorem esumfzf 27743
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 1683 . . . . . 6  |-  F/ k  i  =  1
2 oveq2 6292 . . . . . 6  |-  ( i  =  1  ->  (
1 ... i )  =  ( 1 ... 1
) )
31, 2esumeq1d 27716 . . . . 5  |-  ( i  =  1  -> Σ* k  e.  ( 1 ... i ) ( F `  k
)  = Σ* k  e.  ( 1 ... 1 ) ( F `  k
) )
4 fveq2 5866 . . . . 5  |-  ( i  =  1  ->  (  seq 1 ( +e ,  F ) `  i
)  =  (  seq 1 ( +e ,  F ) `  1
) )
53, 4eqeq12d 2489 . . . 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 316 . . 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 1683 . . . . . 6  |-  F/ k  i  =  n
8 oveq2 6292 . . . . . 6  |-  ( i  =  n  ->  (
1 ... i )  =  ( 1 ... n
) )
97, 8esumeq1d 27716 . . . . 5  |-  ( i  =  n  -> Σ* k  e.  ( 1 ... i ) ( F `  k
)  = Σ* k  e.  ( 1 ... n ) ( F `  k
) )
10 fveq2 5866 . . . . 5  |-  ( i  =  n  ->  (  seq 1 ( +e ,  F ) `  i
)  =  (  seq 1 ( +e ,  F ) `  n
) )
119, 10eqeq12d 2489 . . . 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 316 . . 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 1683 . . . . . 6  |-  F/ k  i  =  ( n  +  1 )
14 oveq2 6292 . . . . . 6  |-  ( i  =  ( n  + 
1 )  ->  (
1 ... i )  =  ( 1 ... (
n  +  1 ) ) )
1513, 14esumeq1d 27716 . . . . 5  |-  ( i  =  ( n  + 
1 )  -> Σ* k  e.  ( 1 ... i ) ( F `  k
)  = Σ* k  e.  ( 1 ... ( n  +  1 ) ) ( F `  k
) )
16 fveq2 5866 . . . . 5  |-  ( i  =  ( n  + 
1 )  ->  (  seq 1 ( +e ,  F ) `  i
)  =  (  seq 1 ( +e ,  F ) `  (
n  +  1 ) ) )
1715, 16eqeq12d 2489 . . . 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 316 . . 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 1683 . . . . . 6  |-  F/ k  i  =  N
20 oveq2 6292 . . . . . 6  |-  ( i  =  N  ->  (
1 ... i )  =  ( 1 ... N
) )
2119, 20esumeq1d 27716 . . . . 5  |-  ( i  =  N  -> Σ* k  e.  ( 1 ... i ) ( F `  k
)  = Σ* k  e.  ( 1 ... N ) ( F `  k
) )
22 fveq2 5866 . . . . 5  |-  ( i  =  N  ->  (  seq 1 ( +e ,  F ) `  i
)  =  (  seq 1 ( +e ,  F ) `  N
) )
2321, 22eqeq12d 2489 . . . 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 316 . . 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 5866 . . . . . 6  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
26 nfcv 2629 . . . . . 6  |-  F/_ x { 1 }
27 nfcv 2629 . . . . . 6  |-  F/_ k { 1 }
28 nfcv 2629 . . . . . 6  |-  F/_ x
( F `  k
)
29 esumfzf.1 . . . . . . 7  |-  F/_ k F
30 nfcv 2629 . . . . . . 7  |-  F/_ k
x
3129, 30nffv 5873 . . . . . 6  |-  F/_ k
( F `  x
)
3225, 26, 27, 28, 31cbvesum 27722 . . . . 5  |- Σ* k  e.  {
1 }  ( F `
 k )  = Σ* x  e.  { 1 }  ( F `  x
)
33 simpr 461 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,] +oo )  /\  x  =  1 )  ->  x  =  1 )
3433fveq2d 5870 . . . . . 6  |-  ( ( F : NN --> ( 0 [,] +oo )  /\  x  =  1 )  ->  ( F `  x )  =  ( F `  1 ) )
35 1z 10894 . . . . . . 7  |-  1  e.  ZZ
3635a1i 11 . . . . . 6  |-  ( F : NN --> ( 0 [,] +oo )  -> 
1  e.  ZZ )
37 1nn 10547 . . . . . . 7  |-  1  e.  NN
38 ffvelrn 6019 . . . . . . 7  |-  ( ( F : NN --> ( 0 [,] +oo )  /\  1  e.  NN )  ->  ( F `  1
)  e.  ( 0 [,] +oo ) )
3937, 38mpan2 671 . . . . . 6  |-  ( F : NN --> ( 0 [,] +oo )  -> 
( F `  1
)  e.  ( 0 [,] +oo ) )
4034, 36, 39esumsn 27740 . . . . 5  |-  ( F : NN --> ( 0 [,] +oo )  -> Σ* x  e.  { 1 }  ( F `  x )  =  ( F ` 
1 ) )
4132, 40syl5eq 2520 . . . 4  |-  ( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  { 1 }  ( F `  k )  =  ( F ` 
1 ) )
42 fzsn 11725 . . . . . 6  |-  ( 1  e.  ZZ  ->  (
1 ... 1 )  =  { 1 } )
4335, 42ax-mp 5 . . . . 5  |-  ( 1 ... 1 )  =  { 1 }
44 esumeq1 27715 . . . . 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 12088 . . . . 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 2533 . . 3  |-  ( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... 1
) ( F `  k )  =  (  seq 1 ( +e ,  F ) `
 1 ) )
49 simpl 457 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  n  e.  NN )
50 nnuz 11117 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
5149, 50syl6eleq 2565 . . . . . . . 8  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  n  e.  (
ZZ>= `  1 ) )
52 seqp1 12090 . . . . . . . 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 465 . . . . . 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 461 . . . . . . 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 6299 . . . . . 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 1683 . . . . . . . . . 10  |-  F/ k  n  e.  NN
5857nfci 2618 . . . . . . . . . . 11  |-  F/_ k NN
59 nfcv 2629 . . . . . . . . . . 11  |-  F/_ k
( 0 [,] +oo )
6029, 58, 59nff 5727 . . . . . . . . . 10  |-  F/ k  F : NN --> ( 0 [,] +oo )
6157, 60nfan 1875 . . . . . . . . 9  |-  F/ k ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )
62 fzsuc 11727 . . . . . . . . . 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 27716 . . . . . . . 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 2629 . . . . . . . . 9  |-  F/_ k
( 1 ... n
)
66 nfcv 2629 . . . . . . . . 9  |-  F/_ k { ( n  + 
1 ) }
67 ovex 6309 . . . . . . . . . 10  |-  ( 1 ... n )  e. 
_V
6867a1i 11 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  ( 1 ... n )  e.  _V )
69 snex 4688 . . . . . . . . . 10  |-  { ( n  +  1 ) }  e.  _V
7069a1i 11 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  { ( n  +  1 ) }  e.  _V )
71 fzp1disj 11738 . . . . . . . . . 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 754 . . . . . . . . . 10  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  ( 1 ... n ) )  ->  F : NN
--> ( 0 [,] +oo ) )
74 fzssnn 27291 . . . . . . . . . . . 12  |-  ( 1  e.  NN  ->  (
1 ... n )  C_  NN )
7537, 74ax-mp 5 . . . . . . . . . . 11  |-  ( 1 ... n )  C_  NN
76 simpr 461 . . . . . . . . . . 11  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  ( 1 ... n ) )  ->  k  e.  ( 1 ... n
) )
7775, 76sseldi 3502 . . . . . . . . . 10  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  ( 1 ... n ) )  ->  k  e.  NN )
7873, 77ffvelrnd 6022 . . . . . . . . 9  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  ( 1 ... n ) )  ->  ( F `  k )  e.  ( 0 [,] +oo )
)
79 simplr 754 . . . . . . . . . 10  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  {
( n  +  1 ) } )  ->  F : NN --> ( 0 [,] +oo ) )
80 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  {
( n  +  1 ) } )  -> 
k  e.  { ( n  +  1 ) } )
81 elsn 4041 . . . . . . . . . . . 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 753 . . . . . . . . . . . 12  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  {
( n  +  1 ) } )  ->  n  e.  NN )
8483peano2nnd 10553 . . . . . . . . . . 11  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  {
( n  +  1 ) } )  -> 
( n  +  1 )  e.  NN )
8582, 84eqeltrd 2555 . . . . . . . . . 10  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  k  e.  {
( n  +  1 ) } )  -> 
k  e.  NN )
8679, 85ffvelrnd 6022 . . . . . . . . 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 27731 . . . . . . . 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 2629 . . . . . . . . . . 11  |-  F/_ x { ( n  + 
1 ) }
8925, 88, 66, 28, 31cbvesum 27722 . . . . . . . . . 10  |- Σ* k  e.  {
( n  +  1 ) }  ( F `
 k )  = Σ* x  e.  { ( n  +  1 ) }  ( F `  x
)
90 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  x  =  ( n  +  1 ) )  ->  x  =  ( n  +  1
) )
9190fveq2d 5870 . . . . . . . . . . 11  |-  ( ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  /\  x  =  ( n  +  1 ) )  ->  ( F `  x )  =  ( F `  ( n  +  1 ) ) )
9249peano2nnd 10553 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  ( n  + 
1 )  e.  NN )
93 simpr 461 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  F : NN --> ( 0 [,] +oo ) )
9493, 92ffvelrnd 6022 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  ->  ( F `  ( n  +  1
) )  e.  ( 0 [,] +oo )
)
9591, 92, 94esumsn 27740 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  -> Σ* x  e.  { ( n  +  1 ) }  ( F `  x )  =  ( F `  ( n  +  1 ) ) )
9689, 95syl5eq 2520 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  F : NN --> ( 0 [,] +oo ) )  -> Σ* k  e.  { ( n  +  1 ) }  ( F `  k )  =  ( F `  ( n  +  1 ) ) )
9796oveq2d 6300 . . . . . . . 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 2513 . . . . . . 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 465 . . . . . 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 2515 . . . . 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 604 . . . 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 10554 . 2  |-  ( N  e.  NN  ->  ( F : NN --> ( 0 [,] +oo )  -> Σ* k  e.  ( 1 ... N
) ( F `  k )  =  (  seq 1 ( +e ,  F ) `
 N ) ) )
104103impcom 430 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 369    = wceq 1379    e. wcel 1767   F/_wnfc 2615   _Vcvv 3113    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   -->wf 5584   ` cfv 5588  (class class class)co 6284   0cc0 9492   1c1 9493    + caddc 9495   +oocpnf 9625   NNcn 10536   ZZcz 10864   ZZ>=cuz 11082   +ecxad 11316   [,]cicc 11532   ...cfz 11672    seqcseq 12075  Σ*cesum 27708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-pi 13670  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-ordt 14756  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-ps 15687  df-tsr 15688  df-mnd 15732  df-plusf 15733  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-cntz 16160  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-subrg 17227  df-abv 17266  df-lmod 17314  df-scaf 17315  df-sra 17618  df-rgmod 17619  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-tmd 20334  df-tgp 20335  df-tsms 20388  df-trg 20425  df-xms 20586  df-ms 20587  df-tms 20588  df-nm 20866  df-ngp 20867  df-nrg 20869  df-nlm 20870  df-ii 21144  df-cncf 21145  df-limc 22033  df-dv 22034  df-log 22700  df-esum 27709
This theorem is referenced by:  esumfsup  27744
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