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Theorem gsumfzmptfidmadd 13398
Description: The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.) (Revised by Jim Kingdon, 31-Aug-2025.)
Hypotheses
Ref Expression
gsummptfidmadd.b  |-  B  =  ( Base `  G
)
gsummptfidmadd.p  |-  .+  =  ( +g  `  G )
gsummptfidmadd.g  |-  ( ph  ->  G  e. CMnd )
gsumfzmptfidmadd.m  |-  ( ph  ->  M  e.  ZZ )
gsumfzmptfidmadd.n  |-  ( ph  ->  N  e.  ZZ )
gsumfzmptfidmadd.c  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  C  e.  B )
gsumfzmptfidmadd.d  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  D  e.  B )
gsumfzmptfidmadd.f  |-  F  =  ( x  e.  ( M ... N ) 
|->  C )
gsumfzmptfidmadd.h  |-  H  =  ( x  e.  ( M ... N ) 
|->  D )
Assertion
Ref Expression
gsumfzmptfidmadd  |-  ( ph  ->  ( G  gsumg  ( x  e.  ( M ... N ) 
|->  ( C  .+  D
) ) )  =  ( ( G  gsumg  F ) 
.+  ( G  gsumg  H ) ) )
Distinct variable groups:    x, B    ph, x    x, 
.+    x, M    x, N
Allowed substitution hints:    C( x)    D( x)    F( x)    G( x)    H( x)

Proof of Theorem gsumfzmptfidmadd
Dummy variables  k  p  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 110 . . . 4  |-  ( (
ph  /\  N  <  M )  ->  N  <  M )
21iftrued 3564 . . 3  |-  ( (
ph  /\  N  <  M )  ->  if ( N  <  M ,  ( 0g `  G ) ,  (  seq M
(  .+  ,  (
x  e.  ( M ... N )  |->  ( C  .+  D ) ) ) `  N
) )  =  ( 0g `  G ) )
3 gsummptfidmadd.b . . . . 5  |-  B  =  ( Base `  G
)
4 eqid 2193 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
5 gsummptfidmadd.p . . . . 5  |-  .+  =  ( +g  `  G )
6 gsummptfidmadd.g . . . . 5  |-  ( ph  ->  G  e. CMnd )
7 gsumfzmptfidmadd.m . . . . 5  |-  ( ph  ->  M  e.  ZZ )
8 gsumfzmptfidmadd.n . . . . 5  |-  ( ph  ->  N  e.  ZZ )
96cmnmndd 13367 . . . . . . . 8  |-  ( ph  ->  G  e.  Mnd )
109adantr 276 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  G  e.  Mnd )
11 gsumfzmptfidmadd.c . . . . . . 7  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  C  e.  B )
12 gsumfzmptfidmadd.d . . . . . . 7  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  D  e.  B )
133, 5mndcl 12994 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  C  e.  B  /\  D  e.  B )  ->  ( C  .+  D
)  e.  B )
1410, 11, 12, 13syl3anc 1249 . . . . . 6  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( C  .+  D )  e.  B
)
1514fmpttd 5705 . . . . 5  |-  ( ph  ->  ( x  e.  ( M ... N ) 
|->  ( C  .+  D
) ) : ( M ... N ) --> B )
163, 4, 5, 6, 7, 8, 15gsumfzval 12964 . . . 4  |-  ( ph  ->  ( G  gsumg  ( x  e.  ( M ... N ) 
|->  ( C  .+  D
) ) )  =  if ( N  < 
M ,  ( 0g
`  G ) ,  (  seq M ( 
.+  ,  ( x  e.  ( M ... N )  |->  ( C 
.+  D ) ) ) `  N ) ) )
1716adantr 276 . . 3  |-  ( (
ph  /\  N  <  M )  ->  ( G  gsumg  ( x  e.  ( M ... N )  |->  ( C  .+  D ) ) )  =  if ( N  <  M ,  ( 0g `  G ) ,  (  seq M (  .+  ,  ( x  e.  ( M ... N
)  |->  ( C  .+  D ) ) ) `
 N ) ) )
18 gsumfzmptfidmadd.f . . . . . . . . 9  |-  F  =  ( x  e.  ( M ... N ) 
|->  C )
1911, 18fmptd 5704 . . . . . . . 8  |-  ( ph  ->  F : ( M ... N ) --> B )
203, 4, 5, 6, 7, 8, 19gsumfzval 12964 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  F )  =  if ( N  <  M ,  ( 0g `  G ) ,  (  seq M (  .+  ,  F ) `  N
) ) )
2120adantr 276 . . . . . 6  |-  ( (
ph  /\  N  <  M )  ->  ( G  gsumg  F )  =  if ( N  <  M , 
( 0g `  G
) ,  (  seq M (  .+  ,  F ) `  N
) ) )
221iftrued 3564 . . . . . 6  |-  ( (
ph  /\  N  <  M )  ->  if ( N  <  M ,  ( 0g `  G ) ,  (  seq M
(  .+  ,  F
) `  N )
)  =  ( 0g
`  G ) )
2321, 22eqtrd 2226 . . . . 5  |-  ( (
ph  /\  N  <  M )  ->  ( G  gsumg  F )  =  ( 0g
`  G ) )
24 gsumfzmptfidmadd.h . . . . . . . . 9  |-  H  =  ( x  e.  ( M ... N ) 
|->  D )
2512, 24fmptd 5704 . . . . . . . 8  |-  ( ph  ->  H : ( M ... N ) --> B )
263, 4, 5, 6, 7, 8, 25gsumfzval 12964 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  H )  =  if ( N  <  M ,  ( 0g `  G ) ,  (  seq M (  .+  ,  H ) `  N
) ) )
2726adantr 276 . . . . . 6  |-  ( (
ph  /\  N  <  M )  ->  ( G  gsumg  H )  =  if ( N  <  M , 
( 0g `  G
) ,  (  seq M (  .+  ,  H ) `  N
) ) )
281iftrued 3564 . . . . . 6  |-  ( (
ph  /\  N  <  M )  ->  if ( N  <  M ,  ( 0g `  G ) ,  (  seq M
(  .+  ,  H
) `  N )
)  =  ( 0g
`  G ) )
2927, 28eqtrd 2226 . . . . 5  |-  ( (
ph  /\  N  <  M )  ->  ( G  gsumg  H )  =  ( 0g
`  G ) )
3023, 29oveq12d 5928 . . . 4  |-  ( (
ph  /\  N  <  M )  ->  ( ( G  gsumg  F )  .+  ( G  gsumg  H ) )  =  ( ( 0g `  G )  .+  ( 0g `  G ) ) )
313, 4mndidcl 13001 . . . . . 6  |-  ( G  e.  Mnd  ->  ( 0g `  G )  e.  B )
323, 5, 4mndlid 13006 . . . . . 6  |-  ( ( G  e.  Mnd  /\  ( 0g `  G )  e.  B )  -> 
( ( 0g `  G )  .+  ( 0g `  G ) )  =  ( 0g `  G ) )
339, 31, 32syl2anc2 412 . . . . 5  |-  ( ph  ->  ( ( 0g `  G )  .+  ( 0g `  G ) )  =  ( 0g `  G ) )
3433adantr 276 . . . 4  |-  ( (
ph  /\  N  <  M )  ->  ( ( 0g `  G )  .+  ( 0g `  G ) )  =  ( 0g
`  G ) )
3530, 34eqtrd 2226 . . 3  |-  ( (
ph  /\  N  <  M )  ->  ( ( G  gsumg  F )  .+  ( G  gsumg  H ) )  =  ( 0g `  G
) )
362, 17, 353eqtr4d 2236 . 2  |-  ( (
ph  /\  N  <  M )  ->  ( G  gsumg  ( x  e.  ( M ... N )  |->  ( C  .+  D ) ) )  =  ( ( G  gsumg  F )  .+  ( G  gsumg  H ) ) )
379ad2antrr 488 . . . . 5  |-  ( ( ( ph  /\  -.  N  <  M )  /\  ( p  e.  B  /\  q  e.  B
) )  ->  G  e.  Mnd )
38 simprl 529 . . . . 5  |-  ( ( ( ph  /\  -.  N  <  M )  /\  ( p  e.  B  /\  q  e.  B
) )  ->  p  e.  B )
39 simprr 531 . . . . 5  |-  ( ( ( ph  /\  -.  N  <  M )  /\  ( p  e.  B  /\  q  e.  B
) )  ->  q  e.  B )
403, 5mndcl 12994 . . . . 5  |-  ( ( G  e.  Mnd  /\  p  e.  B  /\  q  e.  B )  ->  ( p  .+  q
)  e.  B )
4137, 38, 39, 40syl3anc 1249 . . . 4  |-  ( ( ( ph  /\  -.  N  <  M )  /\  ( p  e.  B  /\  q  e.  B
) )  ->  (
p  .+  q )  e.  B )
426ad2antrr 488 . . . . 5  |-  ( ( ( ph  /\  -.  N  <  M )  /\  ( p  e.  B  /\  q  e.  B
) )  ->  G  e. CMnd )
433, 5cmncom 13361 . . . . 5  |-  ( ( G  e. CMnd  /\  p  e.  B  /\  q  e.  B )  ->  (
p  .+  q )  =  ( q  .+  p ) )
4442, 38, 39, 43syl3anc 1249 . . . 4  |-  ( ( ( ph  /\  -.  N  <  M )  /\  ( p  e.  B  /\  q  e.  B
) )  ->  (
p  .+  q )  =  ( q  .+  p ) )
459ad2antrr 488 . . . . 5  |-  ( ( ( ph  /\  -.  N  <  M )  /\  ( p  e.  B  /\  q  e.  B  /\  r  e.  B
) )  ->  G  e.  Mnd )
463, 5mndass 12995 . . . . 5  |-  ( ( G  e.  Mnd  /\  ( p  e.  B  /\  q  e.  B  /\  r  e.  B
) )  ->  (
( p  .+  q
)  .+  r )  =  ( p  .+  ( q  .+  r
) ) )
4745, 46sylancom 420 . . . 4  |-  ( ( ( ph  /\  -.  N  <  M )  /\  ( p  e.  B  /\  q  e.  B  /\  r  e.  B
) )  ->  (
( p  .+  q
)  .+  r )  =  ( p  .+  ( q  .+  r
) ) )
487adantr 276 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  M  e.  ZZ )
498adantr 276 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  N  e.  ZZ )
5048zred 9429 . . . . . 6  |-  ( (
ph  /\  -.  N  <  M )  ->  M  e.  RR )
5149zred 9429 . . . . . 6  |-  ( (
ph  /\  -.  N  <  M )  ->  N  e.  RR )
52 simpr 110 . . . . . 6  |-  ( (
ph  /\  -.  N  <  M )  ->  -.  N  <  M )
5350, 51, 52nltled 8130 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  M  <_  N )
54 eluz2 9588 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N ) )
5548, 49, 53, 54syl3anbrc 1183 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  N  e.  ( ZZ>= `  M )
)
5619adantr 276 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  F : ( M ... N ) --> B )
5756ffvelcdmda 5685 . . . 4  |-  ( ( ( ph  /\  -.  N  <  M )  /\  k  e.  ( M ... N ) )  -> 
( F `  k
)  e.  B )
5825adantr 276 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  H : ( M ... N ) --> B )
5958ffvelcdmda 5685 . . . 4  |-  ( ( ( ph  /\  -.  N  <  M )  /\  k  e.  ( M ... N ) )  -> 
( H `  k
)  e.  B )
607, 8fzfigd 10492 . . . . . . . 8  |-  ( ph  ->  ( M ... N
)  e.  Fin )
6118a1i 9 . . . . . . . 8  |-  ( ph  ->  F  =  ( x  e.  ( M ... N )  |->  C ) )
6224a1i 9 . . . . . . . 8  |-  ( ph  ->  H  =  ( x  e.  ( M ... N )  |->  D ) )
6360, 11, 12, 61, 62offval2 6138 . . . . . . 7  |-  ( ph  ->  ( F  oF  .+  H )  =  ( x  e.  ( M ... N ) 
|->  ( C  .+  D
) ) )
6463fveq1d 5548 . . . . . 6  |-  ( ph  ->  ( ( F  oF  .+  H ) `  k )  =  ( ( x  e.  ( M ... N ) 
|->  ( C  .+  D
) ) `  k
) )
6564ad2antrr 488 . . . . 5  |-  ( ( ( ph  /\  -.  N  <  M )  /\  k  e.  ( M ... N ) )  -> 
( ( F  oF  .+  H ) `  k )  =  ( ( x  e.  ( M ... N ) 
|->  ( C  .+  D
) ) `  k
) )
6619ffnd 5396 . . . . . . 7  |-  ( ph  ->  F  Fn  ( M ... N ) )
6725ffnd 5396 . . . . . . 7  |-  ( ph  ->  H  Fn  ( M ... N ) )
68 inidm 3368 . . . . . . 7  |-  ( ( M ... N )  i^i  ( M ... N ) )  =  ( M ... N
)
69 eqidd 2194 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  =  ( F `  k ) )
70 eqidd 2194 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  =  ( H `  k ) )
719adantr 276 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  G  e.  Mnd )
7219ffvelcdmda 5685 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  B
)
7325ffvelcdmda 5685 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  e.  B
)
743, 5mndcl 12994 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( F `  k )  e.  B  /\  ( H `  k )  e.  B )  ->  (
( F `  k
)  .+  ( H `  k ) )  e.  B )
7571, 72, 73, 74syl3anc 1249 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( ( F `  k )  .+  ( H `  k
) )  e.  B
)
7666, 67, 60, 60, 68, 69, 70, 75ofvalg 6132 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( ( F  oF  .+  H
) `  k )  =  ( ( F `
 k )  .+  ( H `  k ) ) )
7776adantlr 477 . . . . 5  |-  ( ( ( ph  /\  -.  N  <  M )  /\  k  e.  ( M ... N ) )  -> 
( ( F  oF  .+  H ) `  k )  =  ( ( F `  k
)  .+  ( H `  k ) ) )
7865, 77eqtr3d 2228 . . . 4  |-  ( ( ( ph  /\  -.  N  <  M )  /\  k  e.  ( M ... N ) )  -> 
( ( x  e.  ( M ... N
)  |->  ( C  .+  D ) ) `  k )  =  ( ( F `  k
)  .+  ( H `  k ) ) )
79 plusgslid 12720 . . . . . . . 8  |-  ( +g  = Slot  ( +g  `  ndx )  /\  ( +g  `  ndx )  e.  NN )
8079slotex 12635 . . . . . . 7  |-  ( G  e. CMnd  ->  ( +g  `  G
)  e.  _V )
816, 80syl 14 . . . . . 6  |-  ( ph  ->  ( +g  `  G
)  e.  _V )
825, 81eqeltrid 2280 . . . . 5  |-  ( ph  ->  .+  e.  _V )
8382adantr 276 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  .+  e.  _V )
8419, 60fexd 5780 . . . . 5  |-  ( ph  ->  F  e.  _V )
8584adantr 276 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  F  e.  _V )
8625, 60fexd 5780 . . . . 5  |-  ( ph  ->  H  e.  _V )
8786adantr 276 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  H  e.  _V )
8815, 60fexd 5780 . . . . 5  |-  ( ph  ->  ( x  e.  ( M ... N ) 
|->  ( C  .+  D
) )  e.  _V )
8988adantr 276 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  (
x  e.  ( M ... N )  |->  ( C  .+  D ) )  e.  _V )
9041, 44, 47, 55, 57, 59, 78, 83, 85, 87, 89seqcaoprg 10557 . . 3  |-  ( (
ph  /\  -.  N  <  M )  ->  (  seq M (  .+  , 
( x  e.  ( M ... N ) 
|->  ( C  .+  D
) ) ) `  N )  =  ( (  seq M ( 
.+  ,  F ) `
 N )  .+  (  seq M (  .+  ,  H ) `  N
) ) )
9116adantr 276 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  ( G  gsumg  ( x  e.  ( M ... N ) 
|->  ( C  .+  D
) ) )  =  if ( N  < 
M ,  ( 0g
`  G ) ,  (  seq M ( 
.+  ,  ( x  e.  ( M ... N )  |->  ( C 
.+  D ) ) ) `  N ) ) )
9252iffalsed 3567 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  if ( N  <  M , 
( 0g `  G
) ,  (  seq M (  .+  , 
( x  e.  ( M ... N ) 
|->  ( C  .+  D
) ) ) `  N ) )  =  (  seq M ( 
.+  ,  ( x  e.  ( M ... N )  |->  ( C 
.+  D ) ) ) `  N ) )
9391, 92eqtrd 2226 . . 3  |-  ( (
ph  /\  -.  N  <  M )  ->  ( G  gsumg  ( x  e.  ( M ... N ) 
|->  ( C  .+  D
) ) )  =  (  seq M ( 
.+  ,  ( x  e.  ( M ... N )  |->  ( C 
.+  D ) ) ) `  N ) )
9420adantr 276 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  ( G  gsumg  F )  =  if ( N  <  M ,  ( 0g `  G ) ,  (  seq M (  .+  ,  F ) `  N
) ) )
9552iffalsed 3567 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  if ( N  <  M , 
( 0g `  G
) ,  (  seq M (  .+  ,  F ) `  N
) )  =  (  seq M (  .+  ,  F ) `  N
) )
9694, 95eqtrd 2226 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  ( G  gsumg  F )  =  (  seq M (  .+  ,  F ) `  N
) )
9726adantr 276 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  ( G  gsumg  H )  =  if ( N  <  M ,  ( 0g `  G ) ,  (  seq M (  .+  ,  H ) `  N
) ) )
9852iffalsed 3567 . . . . 5  |-  ( (
ph  /\  -.  N  <  M )  ->  if ( N  <  M , 
( 0g `  G
) ,  (  seq M (  .+  ,  H ) `  N
) )  =  (  seq M (  .+  ,  H ) `  N
) )
9997, 98eqtrd 2226 . . . 4  |-  ( (
ph  /\  -.  N  <  M )  ->  ( G  gsumg  H )  =  (  seq M (  .+  ,  H ) `  N
) )
10096, 99oveq12d 5928 . . 3  |-  ( (
ph  /\  -.  N  <  M )  ->  (
( G  gsumg  F )  .+  ( G  gsumg  H ) )  =  ( (  seq M
(  .+  ,  F
) `  N )  .+  (  seq M ( 
.+  ,  H ) `
 N ) ) )
10190, 93, 1003eqtr4d 2236 . 2  |-  ( (
ph  /\  -.  N  <  M )  ->  ( G  gsumg  ( x  e.  ( M ... N ) 
|->  ( C  .+  D
) ) )  =  ( ( G  gsumg  F ) 
.+  ( G  gsumg  H ) ) )
102 zdclt 9384 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  -> DECID  N  <  M )
1038, 7, 102syl2anc 411 . . 3  |-  ( ph  -> DECID  N  <  M )
104 exmiddc 837 . . 3  |-  (DECID  N  < 
M  ->  ( N  <  M  \/  -.  N  <  M ) )
105103, 104syl 14 . 2  |-  ( ph  ->  ( N  <  M  \/  -.  N  <  M
) )
10636, 101, 105mpjaodan 799 1  |-  ( ph  ->  ( G  gsumg  ( x  e.  ( M ... N ) 
|->  ( C  .+  D
) ) )  =  ( ( G  gsumg  F ) 
.+  ( G  gsumg  H ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 709  DECID wdc 835    /\ w3a 980    = wceq 1364    e. wcel 2164   _Vcvv 2760   ifcif 3557   class class class wbr 4029    |-> cmpt 4090   -->wf 5242   ` cfv 5246  (class class class)co 5910    oFcof 6120   Fincfn 6785    < clt 8044    <_ cle 8045   ZZcz 9307   ZZ>=cuz 9582   ...cfz 10064    seqcseq 10508   Basecbs 12608   +g cplusg 12685   0gc0g 12857    gsumg cgsu 12858   Mndcmnd 12987  CMndccmn 13343
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-rmo 2480  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-of 6122  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-2 9031  df-n0 9231  df-z 9308  df-uz 9583  df-fz 10065  df-fzo 10199  df-seqfrec 10509  df-ndx 12611  df-slot 12612  df-base 12614  df-plusg 12698  df-0g 12859  df-igsum 12860  df-mgm 12929  df-sgrp 12975  df-mnd 12988  df-cmn 13345
This theorem is referenced by:  gsumfzmptfidmadd2  13399
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