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Theorem gsumzsplit 15521
Description: Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
gsumzsplit.b  |-  B  =  ( Base `  G
)
gsumzsplit.0  |-  .0.  =  ( 0g `  G )
gsumzsplit.p  |-  .+  =  ( +g  `  G )
gsumzsplit.z  |-  Z  =  (Cntz `  G )
gsumzsplit.g  |-  ( ph  ->  G  e.  Mnd )
gsumzsplit.a  |-  ( ph  ->  A  e.  V )
gsumzsplit.f  |-  ( ph  ->  F : A --> B )
gsumzsplit.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzsplit.w  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
gsumzsplit.i  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
gsumzsplit.u  |-  ( ph  ->  A  =  ( C  u.  D ) )
Assertion
Ref Expression
gsumzsplit  |-  ( ph  ->  ( G  gsumg  F )  =  ( ( G  gsumg  ( F  |`  C ) )  .+  ( G 
gsumg  ( F  |`  D ) ) ) )

Proof of Theorem gsumzsplit
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 gsumzsplit.b . . 3  |-  B  =  ( Base `  G
)
2 gsumzsplit.0 . . 3  |-  .0.  =  ( 0g `  G )
3 gsumzsplit.p . . 3  |-  .+  =  ( +g  `  G )
4 gsumzsplit.z . . 3  |-  Z  =  (Cntz `  G )
5 gsumzsplit.g . . 3  |-  ( ph  ->  G  e.  Mnd )
6 gsumzsplit.a . . 3  |-  ( ph  ->  A  e.  V )
7 gsumzsplit.w . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
8 gsumzsplit.f . . . . . . . 8  |-  ( ph  ->  F : A --> B )
9 ssid 3359 . . . . . . . . 9  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
109a1i 11 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
118, 10suppssr 5856 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F `  k )  =  .0.  )
1211ifeq1d 3745 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  if ( k  e.  C ,  .0.  ,  .0.  ) )
13 ifid 3763 . . . . . 6  |-  if ( k  e.  C ,  .0.  ,  .0.  )  =  .0.
1412, 13syl6eq 2483 . . . . 5  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  .0.  )
1514suppss2 6292 . . . 4  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
16 ssfi 7321 . . . 4  |-  ( ( ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )  -> 
( `' ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
177, 15, 16syl2anc 643 . . 3  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
1811ifeq1d 3745 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  if ( k  e.  D ,  .0.  ,  .0.  ) )
19 ifid 3763 . . . . . 6  |-  if ( k  e.  D ,  .0.  ,  .0.  )  =  .0.
2018, 19syl6eq 2483 . . . . 5  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  .0.  )
2120suppss2 6292 . . . 4  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
22 ssfi 7321 . . . 4  |-  ( ( ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )  -> 
( `' ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
237, 21, 22syl2anc 643 . . 3  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
241submacs 14757 . . . . 5  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
25 acsmre 13869 . . . . 5  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
265, 24, 253syl 19 . . . 4  |-  ( ph  ->  (SubMnd `  G )  e.  (Moore `  B )
)
27 frn 5589 . . . . 5  |-  ( F : A --> B  ->  ran  F  C_  B )
288, 27syl 16 . . . 4  |-  ( ph  ->  ran  F  C_  B
)
29 eqid 2435 . . . . 5  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
3029mrccl 13828 . . . 4  |-  ( ( (SubMnd `  G )  e.  (Moore `  B )  /\  ran  F  C_  B
)  ->  ( (mrCls `  (SubMnd `  G )
) `  ran  F )  e.  (SubMnd `  G
) )
3126, 28, 30syl2anc 643 . . 3  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  e.  (SubMnd `  G ) )
32 gsumzsplit.c . . . . 5  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
33 eqid 2435 . . . . . 6  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
344, 29, 33cntzspan 15452 . . . . 5  |-  ( ( G  e.  Mnd  /\  ran  F  C_  ( Z `  ran  F ) )  ->  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  e. CMnd )
355, 32, 34syl2anc 643 . . . 4  |-  ( ph  ->  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)  e. CMnd )
3633, 4submcmn2 15450 . . . . 5  |-  ( ( (mrCls `  (SubMnd `  G
) ) `  ran  F )  e.  (SubMnd `  G )  ->  (
( Gs  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)  e. CMnd  <->  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )  C_  ( Z `  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) ) ) )
3731, 36syl 16 . . . 4  |-  ( ph  ->  ( ( Gs  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  e. CMnd  <->  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) 
C_  ( Z `  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) ) ) )
3835, 37mpbid 202 . . 3  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  C_  ( Z `  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) ) )
3926, 29, 28mrcssidd 13842 . . . . . . 7  |-  ( ph  ->  ran  F  C_  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) )
4039adantr 452 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ran  F 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
41 ffn 5583 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
428, 41syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  A )
43 fnfvelrn 5859 . . . . . . 7  |-  ( ( F  Fn  A  /\  k  e.  A )  ->  ( F `  k
)  e.  ran  F
)
4442, 43sylan 458 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  ran  F )
4540, 44sseldd 3341 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
462subm0cl 14744 . . . . . . 7  |-  ( ( (mrCls `  (SubMnd `  G
) ) `  ran  F )  e.  (SubMnd `  G )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
4731, 46syl 16 . . . . . 6  |-  ( ph  ->  .0.  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
4847adantr 452 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
49 ifcl 3767 . . . . 5  |-  ( ( ( F `  k
)  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F )  /\  .0.  e.  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
5045, 48, 49syl2anc 643 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
51 eqid 2435 . . . 4  |-  ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)
5250, 51fmptd 5885 . . 3  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) : A --> ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
53 ifcl 3767 . . . . 5  |-  ( ( ( F `  k
)  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F )  /\  .0.  e.  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
5445, 48, 53syl2anc 643 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
55 eqid 2435 . . . 4  |-  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)
5654, 55fmptd 5885 . . 3  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) : A --> ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
571, 2, 3, 4, 5, 6, 17, 23, 31, 38, 52, 56gsumzadd 15519 . 2  |-  ( ph  ->  ( G  gsumg  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  o F  .+  (
k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) )  =  ( ( G 
gsumg  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) ) 
.+  ( G  gsumg  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) ) ) )
588feqmptd 5771 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
59 iftrue 3737 . . . . . . . . . 10  |-  ( k  e.  C  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
6059adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
61 gsumzsplit.i . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
62 noel 3624 . . . . . . . . . . . . . . . 16  |-  -.  k  e.  (/)
63 eleq2 2496 . . . . . . . . . . . . . . . 16  |-  ( ( C  i^i  D )  =  (/)  ->  ( k  e.  ( C  i^i  D )  <->  k  e.  (/) ) )
6462, 63mtbiri 295 . . . . . . . . . . . . . . 15  |-  ( ( C  i^i  D )  =  (/)  ->  -.  k  e.  ( C  i^i  D
) )
6561, 64syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  k  e.  ( C  i^i  D ) )
6665adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  A )  ->  -.  k  e.  ( C  i^i  D ) )
67 elin 3522 . . . . . . . . . . . . 13  |-  ( k  e.  ( C  i^i  D )  <->  ( k  e.  C  /\  k  e.  D ) )
6866, 67sylnib 296 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  A )  ->  -.  ( k  e.  C  /\  k  e.  D
) )
69 imnan 412 . . . . . . . . . . . 12  |-  ( ( k  e.  C  ->  -.  k  e.  D
)  <->  -.  ( k  e.  C  /\  k  e.  D ) )
7068, 69sylibr 204 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  (
k  e.  C  ->  -.  k  e.  D
) )
7170imp 419 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  -.  k  e.  D )
72 iffalse 3738 . . . . . . . . . 10  |-  ( -.  k  e.  D  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  .0.  )
7371, 72syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  .0.  )
7460, 73oveq12d 6091 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  =  ( ( F `
 k )  .+  .0.  ) )
758ffvelrnda 5862 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  B )
761, 3, 2mndrid 14709 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  ( F `  k )  e.  B )  -> 
( ( F `  k )  .+  .0.  )  =  ( F `  k ) )
775, 76sylan 458 . . . . . . . . . 10  |-  ( (
ph  /\  ( F `  k )  e.  B
)  ->  ( ( F `  k )  .+  .0.  )  =  ( F `  k ) )
7875, 77syldan 457 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  .+  .0.  )  =  ( F `  k ) )
7978adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  (
( F `  k
)  .+  .0.  )  =  ( F `  k ) )
8074, 79eqtrd 2467 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  =  ( F `  k ) )
8170con2d 109 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  (
k  e.  D  ->  -.  k  e.  C
) )
8281imp 419 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  -.  k  e.  C )
83 iffalse 3738 . . . . . . . . . 10  |-  ( -.  k  e.  C  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  .0.  )
8482, 83syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  .0.  )
85 iftrue 3737 . . . . . . . . . 10  |-  ( k  e.  D  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
8685adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
8784, 86oveq12d 6091 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  =  (  .0.  .+  ( F `  k ) ) )
881, 3, 2mndlid 14708 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  ( F `  k )  e.  B )  -> 
(  .0.  .+  ( F `  k )
)  =  ( F `
 k ) )
895, 88sylan 458 . . . . . . . . . 10  |-  ( (
ph  /\  ( F `  k )  e.  B
)  ->  (  .0.  .+  ( F `  k
) )  =  ( F `  k ) )
9075, 89syldan 457 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  (  .0.  .+  ( F `  k ) )  =  ( F `  k
) )
9190adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  (  .0.  .+  ( F `  k ) )  =  ( F `  k
) )
9287, 91eqtrd 2467 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  =  ( F `  k ) )
93 gsumzsplit.u . . . . . . . . . 10  |-  ( ph  ->  A  =  ( C  u.  D ) )
9493eleq2d 2502 . . . . . . . . 9  |-  ( ph  ->  ( k  e.  A  <->  k  e.  ( C  u.  D ) ) )
95 elun 3480 . . . . . . . . 9  |-  ( k  e.  ( C  u.  D )  <->  ( k  e.  C  \/  k  e.  D ) )
9694, 95syl6bb 253 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  <->  ( k  e.  C  \/  k  e.  D )
) )
9796biimpa 471 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  (
k  e.  C  \/  k  e.  D )
)
9880, 92, 97mpjaodan 762 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  =  ( F `  k ) )
9998mpteq2dva 4287 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) )  =  ( k  e.  A  |->  ( F `  k
) ) )
10058, 99eqtr4d 2470 . . . 4  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) )
1011, 2mndidcl 14706 . . . . . . . 8  |-  ( G  e.  Mnd  ->  .0.  e.  B )
1025, 101syl 16 . . . . . . 7  |-  ( ph  ->  .0.  e.  B )
103102adantr 452 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  .0.  e.  B )
104 ifcl 3767 . . . . . 6  |-  ( ( ( F `  k
)  e.  B  /\  .0.  e.  B )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  e.  B )
10575, 103, 104syl2anc 643 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  e.  B )
106 ifcl 3767 . . . . . 6  |-  ( ( ( F `  k
)  e.  B  /\  .0.  e.  B )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  e.  B )
10775, 103, 106syl2anc 643 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  e.  B )
108 eqidd 2436 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) )  =  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) )
109 eqidd 2436 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) )  =  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) )
1106, 105, 107, 108, 109offval2 6314 . . . 4  |-  ( ph  ->  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  o F  .+  (
k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) )  =  ( k  e.  A  |->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) )
111100, 110eqtr4d 2470 . . 3  |-  ( ph  ->  F  =  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  o F 
.+  ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) )
112111oveq2d 6089 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  o F  .+  (
k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) ) )
11358reseq1d 5137 . . . . . 6  |-  ( ph  ->  ( F  |`  C )  =  ( ( k  e.  A  |->  ( F `
 k ) )  |`  C ) )
114 ssun1 3502 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
115114, 93syl5sseqr 3389 . . . . . . 7  |-  ( ph  ->  C  C_  A )
11659mpteq2ia 4283 . . . . . . . 8  |-  ( k  e.  C  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  C  |->  ( F `
 k ) )
117 resmpt 5183 . . . . . . . 8  |-  ( C 
C_  A  ->  (
( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) )  |`  C )  =  ( k  e.  C  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) ) )
118 resmpt 5183 . . . . . . . 8  |-  ( C 
C_  A  ->  (
( k  e.  A  |->  ( F `  k
) )  |`  C )  =  ( k  e.  C  |->  ( F `  k ) ) )
119116, 117, 1183eqtr4a 2493 . . . . . . 7  |-  ( C 
C_  A  ->  (
( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) )  |`  C )  =  ( ( k  e.  A  |->  ( F `  k
) )  |`  C ) )
120115, 119syl 16 . . . . . 6  |-  ( ph  ->  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  |`  C )  =  ( ( k  e.  A  |->  ( F `  k
) )  |`  C ) )
121113, 120eqtr4d 2470 . . . . 5  |-  ( ph  ->  ( F  |`  C )  =  ( ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  |`  C ) )
122121oveq2d 6089 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  =  ( G 
gsumg  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  |`  C ) ) )
123105, 51fmptd 5885 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) : A --> B )
124 frn 5589 . . . . . . 7  |-  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) ) : A --> ( (mrCls `  (SubMnd `  G
) ) `  ran  F )  ->  ran  ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  C_  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
12552, 124syl 16 . . . . . 6  |-  ( ph  ->  ran  ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) ) 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
1264cntzidss 15128 . . . . . 6  |-  ( ( ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  C_  ( Z `  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  /\  ran  (
k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  C_  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) )  ->  ran  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) )  C_  ( Z `  ran  (
k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) ) ) )
12738, 125, 126syl2anc 643 . . . . 5  |-  ( ph  ->  ran  ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) ) 
C_  ( Z `  ran  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) ) )
128 eldifn 3462 . . . . . . . 8  |-  ( k  e.  ( A  \  C )  ->  -.  k  e.  C )
129128adantl 453 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  C ) )  ->  -.  k  e.  C )
130129, 83syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  C ) )  ->  if (
k  e.  C , 
( F `  k
) ,  .0.  )  =  .0.  )
131130suppss2 6292 . . . . 5  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  C
)
1321, 2, 4, 5, 6, 123, 127, 131, 17gsumzres 15509 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  |`  C ) )  =  ( G  gsumg  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) ) )
133122, 132eqtrd 2467 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  =  ( G 
gsumg  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) ) )
13458reseq1d 5137 . . . . . 6  |-  ( ph  ->  ( F  |`  D )  =  ( ( k  e.  A  |->  ( F `
 k ) )  |`  D ) )
135 ssun2 3503 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
136135, 93syl5sseqr 3389 . . . . . . 7  |-  ( ph  ->  D  C_  A )
13785mpteq2ia 4283 . . . . . . . 8  |-  ( k  e.  D  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  D  |->  ( F `
 k ) )
138 resmpt 5183 . . . . . . . 8  |-  ( D 
C_  A  ->  (
( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) )  |`  D )  =  ( k  e.  D  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) )
139 resmpt 5183 . . . . . . . 8  |-  ( D 
C_  A  ->  (
( k  e.  A  |->  ( F `  k
) )  |`  D )  =  ( k  e.  D  |->  ( F `  k ) ) )
140137, 138, 1393eqtr4a 2493 . . . . . . 7  |-  ( D 
C_  A  ->  (
( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) )  |`  D )  =  ( ( k  e.  A  |->  ( F `  k
) )  |`  D ) )
141136, 140syl 16 . . . . . 6  |-  ( ph  ->  ( ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  |`  D )  =  ( ( k  e.  A  |->  ( F `  k
) )  |`  D ) )
142134, 141eqtr4d 2470 . . . . 5  |-  ( ph  ->  ( F  |`  D )  =  ( ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  |`  D ) )
143142oveq2d 6089 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  D ) )  =  ( G 
gsumg  ( ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  |`  D ) ) )
144107, 55fmptd 5885 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) : A --> B )
145 frn 5589 . . . . . . 7  |-  ( ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) : A --> ( (mrCls `  (SubMnd `  G
) ) `  ran  F )  ->  ran  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  C_  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
14656, 145syl 16 . . . . . 6  |-  ( ph  ->  ran  ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
1474cntzidss 15128 . . . . . 6  |-  ( ( ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  C_  ( Z `  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  /\  ran  (
k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  C_  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) )  ->  ran  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) )  C_  ( Z `  ran  (
k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) )
14838, 146, 147syl2anc 643 . . . . 5  |-  ( ph  ->  ran  ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) 
C_  ( Z `  ran  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) ) )
149 eldifn 3462 . . . . . . . 8  |-  ( k  e.  ( A  \  D )  ->  -.  k  e.  D )
150149adantl 453 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  D ) )  ->  -.  k  e.  D )
151150, 72syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  D ) )  ->  if (
k  e.  D , 
( F `  k
) ,  .0.  )  =  .0.  )
152151suppss2 6292 . . . . 5  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  D
)
1531, 2, 4, 5, 6, 144, 148, 152, 23gsumzres 15509 . . . 4  |-  ( ph  ->  ( G  gsumg  ( ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  |`  D ) )  =  ( G  gsumg  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) ) )
154143, 153eqtrd 2467 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  D ) )  =  ( G 
gsumg  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) ) )
155133, 154oveq12d 6091 . 2  |-  ( ph  ->  ( ( G  gsumg  ( F  |`  C ) )  .+  ( G  gsumg  ( F  |`  D ) ) )  =  ( ( G  gsumg  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) ) 
.+  ( G  gsumg  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) ) ) )
15657, 112, 1553eqtr4d 2477 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( ( G  gsumg  ( F  |`  C ) )  .+  ( G 
gsumg  ( F  |`  D ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   ifcif 3731   {csn 3806    e. cmpt 4258   `'ccnv 4869   ran crn 4871    |` cres 4872   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   Fincfn 7101   Basecbs 13461   ↾s cress 13462   +g cplusg 13521   0gc0g 13715    gsumg cgsu 13716  Moorecmre 13799  mrClscmrc 13800  ACScacs 13802   Mndcmnd 14676  SubMndcsubmnd 14729  Cntzccntz 15106  CMndccmn 15404
This theorem is referenced by:  gsumsplit  15522  dpjidcl  15608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-cntz 15108  df-cmn 15406
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