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Theorem gsumfzval 12964
Description: An expression for  gsumg when summing over a finite set of sequential integers. (Contributed by Jim Kingdon, 14-Aug-2025.)
Hypotheses
Ref Expression
gsumval.b  |-  B  =  ( Base `  G
)
gsumval.z  |-  .0.  =  ( 0g `  G )
gsumval.p  |-  .+  =  ( +g  `  G )
gsumval.g  |-  ( ph  ->  G  e.  V )
gsumfzval.m  |-  ( ph  ->  M  e.  ZZ )
gsumfzval.n  |-  ( ph  ->  N  e.  ZZ )
gsumfzval.f  |-  ( ph  ->  F : ( M ... N ) --> B )
Assertion
Ref Expression
gsumfzval  |-  ( ph  ->  ( G  gsumg  F )  =  if ( N  <  M ,  .0.  ,  (  seq M (  .+  ,  F ) `  N
) ) )

Proof of Theorem gsumfzval
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval.b . . 3  |-  B  =  ( Base `  G
)
2 gsumval.z . . 3  |-  .0.  =  ( 0g `  G )
3 gsumval.p . . 3  |-  .+  =  ( +g  `  G )
4 gsumval.g . . 3  |-  ( ph  ->  G  e.  V )
5 gsumfzval.m . . . 4  |-  ( ph  ->  M  e.  ZZ )
6 gsumfzval.n . . . 4  |-  ( ph  ->  N  e.  ZZ )
75, 6fzfigd 10492 . . 3  |-  ( ph  ->  ( M ... N
)  e.  Fin )
8 gsumfzval.f . . 3  |-  ( ph  ->  F : ( M ... N ) --> B )
91, 2, 3, 4, 7, 8igsumval 12963 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  ( iota x ( ( ( M ... N
)  =  (/)  /\  x  =  .0.  )  \/  E. m E. n  e.  (
ZZ>= `  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ) )
10 fn0g 12948 . . . . . 6  |-  0g  Fn  _V
114elexd 2773 . . . . . 6  |-  ( ph  ->  G  e.  _V )
12 funfvex 5563 . . . . . . 7  |-  ( ( Fun  0g  /\  G  e.  dom  0g )  -> 
( 0g `  G
)  e.  _V )
1312funfni 5346 . . . . . 6  |-  ( ( 0g  Fn  _V  /\  G  e.  _V )  ->  ( 0g `  G
)  e.  _V )
1410, 11, 13sylancr 414 . . . . 5  |-  ( ph  ->  ( 0g `  G
)  e.  _V )
152, 14eqeltrid 2280 . . . 4  |-  ( ph  ->  .0.  e.  _V )
16 seqex 10510 . . . . 5  |-  seq M
(  .+  ,  F
)  e.  _V
17 fvexg 5565 . . . . 5  |-  ( (  seq M (  .+  ,  F )  e.  _V  /\  N  e.  ZZ )  ->  (  seq M
(  .+  ,  F
) `  N )  e.  _V )
1816, 6, 17sylancr 414 . . . 4  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  e. 
_V )
1915, 18ifexd 4513 . . 3  |-  ( ph  ->  if ( N  < 
M ,  .0.  , 
(  seq M (  .+  ,  F ) `  N
) )  e.  _V )
20 zdclt 9384 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  -> DECID  N  <  M )
216, 5, 20syl2anc 411 . . . . . . 7  |-  ( ph  -> DECID  N  <  M )
22 eqifdc 3592 . . . . . . 7  |-  (DECID  N  < 
M  ->  ( x  =  if ( N  < 
M ,  .0.  , 
(  seq M (  .+  ,  F ) `  N
) )  <->  ( ( N  <  M  /\  x  =  .0.  )  \/  ( -.  N  <  M  /\  x  =  (  seq M (  .+  ,  F ) `  N
) ) ) ) )
2321, 22syl 14 . . . . . 6  |-  ( ph  ->  ( x  =  if ( N  <  M ,  .0.  ,  (  seq M (  .+  ,  F ) `  N
) )  <->  ( ( N  <  M  /\  x  =  .0.  )  \/  ( -.  N  <  M  /\  x  =  (  seq M (  .+  ,  F ) `  N
) ) ) ) )
24 fzn 10098 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  <->  ( M ... N )  =  (/) ) )
255, 6, 24syl2anc 411 . . . . . . . 8  |-  ( ph  ->  ( N  <  M  <->  ( M ... N )  =  (/) ) )
2625anbi1d 465 . . . . . . 7  |-  ( ph  ->  ( ( N  < 
M  /\  x  =  .0.  )  <->  ( ( M ... N )  =  (/)  /\  x  =  .0.  ) ) )
275adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  M  e.  ZZ )
2827zred 9429 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  M  e.  RR )
296adantr 276 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  N  e.  ZZ )
3029zred 9429 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  N  e.  RR )
31 simprl 529 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  -.  N  <  M )
3228, 30, 31nltled 8130 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  M  <_  N )
33 eluz 9595 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  e.  (
ZZ>= `  M )  <->  M  <_  N ) )
3427, 29, 33syl2anc 411 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  ( N  e.  ( ZZ>= `  M )  <->  M  <_  N ) )
3532, 34mpbird 167 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  N  e.  ( ZZ>= `  M )
)
36 oveq2 5918 . . . . . . . . . . . . . 14  |-  ( n  =  N  ->  ( M ... n )  =  ( M ... N
) )
3736eqeq2d 2205 . . . . . . . . . . . . 13  |-  ( n  =  N  ->  (
( M ... N
)  =  ( M ... n )  <->  ( M ... N )  =  ( M ... N ) ) )
38 fveq2 5546 . . . . . . . . . . . . . 14  |-  ( n  =  N  ->  (  seq M (  .+  ,  F ) `  n
)  =  (  seq M (  .+  ,  F ) `  N
) )
3938eqeq2d 2205 . . . . . . . . . . . . 13  |-  ( n  =  N  ->  (
x  =  (  seq M (  .+  ,  F ) `  n
)  <->  x  =  (  seq M (  .+  ,  F ) `  N
) ) )
4037, 39anbi12d 473 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (
( ( M ... N )  =  ( M ... n )  /\  x  =  (  seq M (  .+  ,  F ) `  n
) )  <->  ( ( M ... N )  =  ( M ... N
)  /\  x  =  (  seq M (  .+  ,  F ) `  N
) ) ) )
4140adantl 277 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( -.  N  <  M  /\  x  =  (  seq M (  .+  ,  F ) `  N
) ) )  /\  n  =  N )  ->  ( ( ( M ... N )  =  ( M ... n
)  /\  x  =  (  seq M (  .+  ,  F ) `  n
) )  <->  ( ( M ... N )  =  ( M ... N
)  /\  x  =  (  seq M (  .+  ,  F ) `  N
) ) ) )
42 eqidd 2194 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  ( M ... N )  =  ( M ... N ) )
43 simprr 531 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  x  =  (  seq M (  .+  ,  F ) `  N
) )
4442, 43jca 306 . . . . . . . . . . 11  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  ( ( M ... N )  =  ( M ... N
)  /\  x  =  (  seq M (  .+  ,  F ) `  N
) ) )
4535, 41, 44rspcedvd 2870 . . . . . . . . . 10  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  E. n  e.  ( ZZ>= `  M )
( ( M ... N )  =  ( M ... n )  /\  x  =  (  seq M (  .+  ,  F ) `  n
) ) )
46 fveq2 5546 . . . . . . . . . . . 12  |-  ( m  =  M  ->  ( ZZ>=
`  m )  =  ( ZZ>= `  M )
)
47 oveq1 5917 . . . . . . . . . . . . . 14  |-  ( m  =  M  ->  (
m ... n )  =  ( M ... n
) )
4847eqeq2d 2205 . . . . . . . . . . . . 13  |-  ( m  =  M  ->  (
( M ... N
)  =  ( m ... n )  <->  ( M ... N )  =  ( M ... n ) ) )
49 seqeq1 10511 . . . . . . . . . . . . . . 15  |-  ( m  =  M  ->  seq m (  .+  ,  F )  =  seq M (  .+  ,  F ) )
5049fveq1d 5548 . . . . . . . . . . . . . 14  |-  ( m  =  M  ->  (  seq m (  .+  ,  F ) `  n
)  =  (  seq M (  .+  ,  F ) `  n
) )
5150eqeq2d 2205 . . . . . . . . . . . . 13  |-  ( m  =  M  ->  (
x  =  (  seq m (  .+  ,  F ) `  n
)  <->  x  =  (  seq M (  .+  ,  F ) `  n
) ) )
5248, 51anbi12d 473 . . . . . . . . . . . 12  |-  ( m  =  M  ->  (
( ( M ... N )  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) )  <->  ( ( M ... N )  =  ( M ... n
)  /\  x  =  (  seq M (  .+  ,  F ) `  n
) ) ) )
5346, 52rexeqbidv 2707 . . . . . . . . . . 11  |-  ( m  =  M  ->  ( E. n  e.  ( ZZ>=
`  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) )  <->  E. n  e.  ( ZZ>= `  M )
( ( M ... N )  =  ( M ... n )  /\  x  =  (  seq M (  .+  ,  F ) `  n
) ) ) )
5453spcegv 2848 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( E. n  e.  ( ZZ>=
`  M ) ( ( M ... N
)  =  ( M ... n )  /\  x  =  (  seq M (  .+  ,  F ) `  n
) )  ->  E. m E. n  e.  ( ZZ>=
`  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) )
5527, 45, 54sylc 62 . . . . . . . . 9  |-  ( (
ph  /\  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) )  ->  E. m E. n  e.  ( ZZ>=
`  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )
5655ex 115 . . . . . . . 8  |-  ( ph  ->  ( ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) )  ->  E. m E. n  e.  ( ZZ>= `  m )
( ( M ... N )  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) )
57 eluzel2 9587 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( ZZ>= `  m
)  ->  m  e.  ZZ )
5857ad2antlr 489 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  ->  m  e.  ZZ )
5958zred 9429 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  ->  m  e.  RR )
60 eluzelre 9592 . . . . . . . . . . . . . 14  |-  ( n  e.  ( ZZ>= `  m
)  ->  n  e.  RR )
6160ad2antlr 489 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  ->  n  e.  RR )
62 eluzle 9594 . . . . . . . . . . . . . 14  |-  ( n  e.  ( ZZ>= `  m
)  ->  m  <_  n )
6362ad2antlr 489 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  ->  m  <_  n )
6459, 61, 63lensymd 8131 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  ->  -.  n  <  m )
65 simprl 529 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  -> 
( M ... N
)  =  ( m ... n ) )
6665eqcomd 2199 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  -> 
( m ... n
)  =  ( M ... N ) )
67 fzopth 10117 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( ZZ>= `  m
)  ->  ( (
m ... n )  =  ( M ... N
)  <->  ( m  =  M  /\  n  =  N ) ) )
6867ad2antlr 489 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  -> 
( ( m ... n )  =  ( M ... N )  <-> 
( m  =  M  /\  n  =  N ) ) )
6966, 68mpbid 147 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  -> 
( m  =  M  /\  n  =  N ) )
7069simprd 114 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  ->  n  =  N )
7169simpld 112 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  ->  m  =  M )
7270, 71breq12d 4042 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  -> 
( n  <  m  <->  N  <  M ) )
7364, 72mtbid 673 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  ->  -.  N  <  M )
74 simprr 531 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  ->  x  =  (  seq m (  .+  ,  F ) `  n
) )
7571seqeq1d 10514 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  ->  seq m (  .+  ,  F )  =  seq M (  .+  ,  F ) )
7675, 70fveq12d 5553 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  -> 
(  seq m (  .+  ,  F ) `  n
)  =  (  seq M (  .+  ,  F ) `  N
) )
7774, 76eqtrd 2226 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  ->  x  =  (  seq M (  .+  ,  F ) `  N
) )
7873, 77jca 306 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  m )
)  /\  ( ( M ... N )  =  ( m ... n
)  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  -> 
( -.  N  < 
M  /\  x  =  (  seq M (  .+  ,  F ) `  N
) ) )
7978rexlimdva2 2614 . . . . . . . . 9  |-  ( ph  ->  ( E. n  e.  ( ZZ>= `  m )
( ( M ... N )  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) )  ->  ( -.  N  <  M  /\  x  =  (  seq M (  .+  ,  F ) `  N
) ) ) )
8079exlimdv 1830 . . . . . . . 8  |-  ( ph  ->  ( E. m E. n  e.  ( ZZ>= `  m ) ( ( M ... N )  =  ( m ... n )  /\  x  =  (  seq m
(  .+  ,  F
) `  n )
)  ->  ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) ) ) )
8156, 80impbid 129 . . . . . . 7  |-  ( ph  ->  ( ( -.  N  <  M  /\  x  =  (  seq M ( 
.+  ,  F ) `
 N ) )  <->  E. m E. n  e.  ( ZZ>= `  m )
( ( M ... N )  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) )
8226, 81orbi12d 794 . . . . . 6  |-  ( ph  ->  ( ( ( N  <  M  /\  x  =  .0.  )  \/  ( -.  N  <  M  /\  x  =  (  seq M (  .+  ,  F ) `  N
) ) )  <->  ( (
( M ... N
)  =  (/)  /\  x  =  .0.  )  \/  E. m E. n  e.  (
ZZ>= `  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ) )
8323, 82bitr2d 189 . . . . 5  |-  ( ph  ->  ( ( ( ( M ... N )  =  (/)  /\  x  =  .0.  )  \/  E. m E. n  e.  (
ZZ>= `  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  <->  x  =  if ( N  <  M ,  .0.  ,  (  seq M (  .+  ,  F ) `  N
) ) ) )
8483adantr 276 . . . 4  |-  ( (
ph  /\  if ( N  <  M ,  .0.  ,  (  seq M ( 
.+  ,  F ) `
 N ) )  e.  _V )  -> 
( ( ( ( M ... N )  =  (/)  /\  x  =  .0.  )  \/  E. m E. n  e.  (
ZZ>= `  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) )  <->  x  =  if ( N  <  M ,  .0.  ,  (  seq M (  .+  ,  F ) `  N
) ) ) )
8584iota5 5228 . . 3  |-  ( (
ph  /\  if ( N  <  M ,  .0.  ,  (  seq M ( 
.+  ,  F ) `
 N ) )  e.  _V )  -> 
( iota x ( ( ( M ... N
)  =  (/)  /\  x  =  .0.  )  \/  E. m E. n  e.  (
ZZ>= `  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) )  =  if ( N  <  M ,  .0.  ,  (  seq M ( 
.+  ,  F ) `
 N ) ) )
8619, 85mpdan 421 . 2  |-  ( ph  ->  ( iota x ( ( ( M ... N )  =  (/)  /\  x  =  .0.  )  \/  E. m E. n  e.  ( ZZ>= `  m )
( ( M ... N )  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) )  =  if ( N  <  M ,  .0.  ,  (  seq M ( 
.+  ,  F ) `
 N ) ) )
879, 86eqtrd 2226 1  |-  ( ph  ->  ( G  gsumg  F )  =  if ( N  <  M ,  .0.  ,  (  seq M (  .+  ,  F ) `  N
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709  DECID wdc 835    = wceq 1364   E.wex 1503    e. wcel 2164   E.wrex 2473   _Vcvv 2760   (/)c0 3446   ifcif 3557   class class class wbr 4029   iotacio 5205    Fn wfn 5241   -->wf 5242   ` cfv 5246  (class class class)co 5910   Fincfn 6785   RRcr 7861    < clt 8044    <_ cle 8045   ZZcz 9307   ZZ>=cuz 9582   ...cfz 10064    seqcseq 10508   Basecbs 12608   +g cplusg 12685   0gc0g 12857    gsumg cgsu 12858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-iinf 4616  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-distr 7966  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-cnre 7973  ax-pre-ltirr 7974  ax-pre-ltwlin 7975  ax-pre-lttrn 7976  ax-pre-apti 7977  ax-pre-ltadd 7978
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4322  df-iord 4395  df-on 4397  df-ilim 4398  df-suc 4400  df-iom 4619  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-1st 6184  df-2nd 6185  df-recs 6349  df-frec 6435  df-1o 6460  df-er 6578  df-en 6786  df-fin 6788  df-pnf 8046  df-mnf 8047  df-xr 8048  df-ltxr 8049  df-le 8050  df-sub 8182  df-neg 8183  df-inn 8973  df-n0 9231  df-z 9308  df-uz 9583  df-fz 10065  df-seqfrec 10509  df-ndx 12611  df-slot 12612  df-base 12614  df-0g 12859  df-igsum 12860
This theorem is referenced by:  gsumfzz  13057  gsumfzcl  13061  gsumfzreidx  13396  gsumfzsubmcl  13397  gsumfzmptfidmadd  13398
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