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Theorem gsumzsplitOLD 16760
Description: Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of gsumzsplit 16759 as of 5-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
gsumzsplitOLD.b  |-  B  =  ( Base `  G
)
gsumzsplitOLD.0  |-  .0.  =  ( 0g `  G )
gsumzsplitOLD.p  |-  .+  =  ( +g  `  G )
gsumzsplitOLD.z  |-  Z  =  (Cntz `  G )
gsumzsplitOLD.g  |-  ( ph  ->  G  e.  Mnd )
gsumzsplitOLD.a  |-  ( ph  ->  A  e.  V )
gsumzsplitOLD.f  |-  ( ph  ->  F : A --> B )
gsumzsplitOLD.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzsplitOLD.w  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
gsumzsplitOLD.i  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
gsumzsplitOLD.u  |-  ( ph  ->  A  =  ( C  u.  D ) )
Assertion
Ref Expression
gsumzsplitOLD  |-  ( ph  ->  ( G  gsumg  F )  =  ( ( G  gsumg  ( F  |`  C ) )  .+  ( G 
gsumg  ( F  |`  D ) ) ) )

Proof of Theorem gsumzsplitOLD
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 gsumzsplitOLD.b . . 3  |-  B  =  ( Base `  G
)
2 gsumzsplitOLD.0 . . 3  |-  .0.  =  ( 0g `  G )
3 gsumzsplitOLD.p . . 3  |-  .+  =  ( +g  `  G )
4 gsumzsplitOLD.z . . 3  |-  Z  =  (Cntz `  G )
5 gsumzsplitOLD.g . . 3  |-  ( ph  ->  G  e.  Mnd )
6 gsumzsplitOLD.a . . 3  |-  ( ph  ->  A  e.  V )
7 gsumzsplitOLD.w . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
8 gsumzsplitOLD.f . . . . . . . 8  |-  ( ph  ->  F : A --> B )
9 ssid 3523 . . . . . . . . 9  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
109a1i 11 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
118, 10suppssrOLD 6016 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F `  k )  =  .0.  )
1211ifeq1d 3957 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  if ( k  e.  C ,  .0.  ,  .0.  ) )
13 ifid 3976 . . . . . 6  |-  if ( k  e.  C ,  .0.  ,  .0.  )  =  .0.
1412, 13syl6eq 2524 . . . . 5  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  .0.  )
1514suppss2OLD 6515 . . . 4  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
16 ssfi 7741 . . . 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 661 . . 3  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
1811ifeq1d 3957 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  if ( k  e.  D ,  .0.  ,  .0.  ) )
19 ifid 3976 . . . . . 6  |-  if ( k  e.  D ,  .0.  ,  .0.  )  =  .0.
2018, 19syl6eq 2524 . . . . 5  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  .0.  )
2120suppss2OLD 6515 . . . 4  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
22 ssfi 7741 . . . 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 661 . . 3  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
241submacs 15818 . . . . 5  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
25 acsmre 14910 . . . . 5  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
265, 24, 253syl 20 . . . 4  |-  ( ph  ->  (SubMnd `  G )  e.  (Moore `  B )
)
27 frn 5737 . . . . 5  |-  ( F : A --> B  ->  ran  F  C_  B )
288, 27syl 16 . . . 4  |-  ( ph  ->  ran  F  C_  B
)
29 eqid 2467 . . . . 5  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
3029mrccl 14869 . . . 4  |-  ( ( (SubMnd `  G )  e.  (Moore `  B )  /\  ran  F  C_  B
)  ->  ( (mrCls `  (SubMnd `  G )
) `  ran  F )  e.  (SubMnd `  G
) )
3126, 28, 30syl2anc 661 . . 3  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  e.  (SubMnd `  G ) )
32 gsumzsplitOLD.c . . . . 5  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
33 eqid 2467 . . . . . 6  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
344, 29, 33cntzspan 16665 . . . . 5  |-  ( ( G  e.  Mnd  /\  ran  F  C_  ( Z `  ran  F ) )  ->  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  e. CMnd )
355, 32, 34syl2anc 661 . . . 4  |-  ( ph  ->  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)  e. CMnd )
3633, 4submcmn2 16662 . . . . 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 210 . . 3  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  C_  ( Z `  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) ) )
3926, 29, 28mrcssidd 14883 . . . . . . 7  |-  ( ph  ->  ran  F  C_  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) )
4039adantr 465 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ran  F 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
41 ffn 5731 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
428, 41syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  A )
43 fnfvelrn 6019 . . . . . . 7  |-  ( ( F  Fn  A  /\  k  e.  A )  ->  ( F `  k
)  e.  ran  F
)
4442, 43sylan 471 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  ran  F )
4540, 44sseldd 3505 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
462subm0cl 15805 . . . . . . 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 465 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
49 ifcl 3981 . . . . 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 661 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
51 eqid 2467 . . . 4  |-  ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)
5250, 51fmptd 6046 . . 3  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) : A --> ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
53 ifcl 3981 . . . . 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 661 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
55 eqid 2467 . . . 4  |-  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)
5654, 55fmptd 6046 . . 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, 56gsumzaddOLD 16752 . 2  |-  ( ph  ->  ( G  gsumg  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  oF  .+  (
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 5921 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
59 iftrue 3945 . . . . . . . . . 10  |-  ( k  e.  C  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
6059adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
61 gsumzsplitOLD.i . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
62 noel 3789 . . . . . . . . . . . . . . . 16  |-  -.  k  e.  (/)
63 eleq2 2540 . . . . . . . . . . . . . . . 16  |-  ( ( C  i^i  D )  =  (/)  ->  ( k  e.  ( C  i^i  D )  <->  k  e.  (/) ) )
6462, 63mtbiri 303 . . . . . . . . . . . . . . 15  |-  ( ( C  i^i  D )  =  (/)  ->  -.  k  e.  ( C  i^i  D
) )
6561, 64syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  k  e.  ( C  i^i  D ) )
6665adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  A )  ->  -.  k  e.  ( C  i^i  D ) )
67 elin 3687 . . . . . . . . . . . . 13  |-  ( k  e.  ( C  i^i  D )  <->  ( k  e.  C  /\  k  e.  D ) )
6866, 67sylnib 304 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  A )  ->  -.  ( k  e.  C  /\  k  e.  D
) )
69 imnan 422 . . . . . . . . . . . 12  |-  ( ( k  e.  C  ->  -.  k  e.  D
)  <->  -.  ( k  e.  C  /\  k  e.  D ) )
7068, 69sylibr 212 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  (
k  e.  C  ->  -.  k  e.  D
) )
7170imp 429 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  -.  k  e.  D )
72 iffalse 3948 . . . . . . . . . 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 6303 . . . . . . . 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 6022 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  B )
761, 3, 2mndrid 15762 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  ( F `  k )  e.  B )  -> 
( ( F `  k )  .+  .0.  )  =  ( F `  k ) )
775, 76sylan 471 . . . . . . . . . 10  |-  ( (
ph  /\  ( F `  k )  e.  B
)  ->  ( ( F `  k )  .+  .0.  )  =  ( F `  k ) )
7875, 77syldan 470 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  .+  .0.  )  =  ( F `  k ) )
7978adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  (
( F `  k
)  .+  .0.  )  =  ( F `  k ) )
8074, 79eqtrd 2508 . . . . . . 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 115 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  (
k  e.  D  ->  -.  k  e.  C
) )
8281imp 429 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  -.  k  e.  C )
83 iffalse 3948 . . . . . . . . . 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 3945 . . . . . . . . . 10  |-  ( k  e.  D  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
8685adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
8784, 86oveq12d 6303 . . . . . . . 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 15761 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  ( F `  k )  e.  B )  -> 
(  .0.  .+  ( F `  k )
)  =  ( F `
 k ) )
895, 88sylan 471 . . . . . . . . . 10  |-  ( (
ph  /\  ( F `  k )  e.  B
)  ->  (  .0.  .+  ( F `  k
) )  =  ( F `  k ) )
9075, 89syldan 470 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  (  .0.  .+  ( F `  k ) )  =  ( F `  k
) )
9190adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  (  .0.  .+  ( F `  k ) )  =  ( F `  k
) )
9287, 91eqtrd 2508 . . . . . . 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 gsumzsplitOLD.u . . . . . . . . . 10  |-  ( ph  ->  A  =  ( C  u.  D ) )
9493eleq2d 2537 . . . . . . . . 9  |-  ( ph  ->  ( k  e.  A  <->  k  e.  ( C  u.  D ) ) )
95 elun 3645 . . . . . . . . 9  |-  ( k  e.  ( C  u.  D )  <->  ( k  e.  C  \/  k  e.  D ) )
9694, 95syl6bb 261 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  <->  ( k  e.  C  \/  k  e.  D )
) )
9796biimpa 484 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  (
k  e.  C  \/  k  e.  D )
)
9880, 92, 97mpjaodan 784 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  =  ( F `  k ) )
9998mpteq2dva 4533 . . . . 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 2511 . . . 4  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) )
1011, 2mndidcl 15759 . . . . . . . 8  |-  ( G  e.  Mnd  ->  .0.  e.  B )
1025, 101syl 16 . . . . . . 7  |-  ( ph  ->  .0.  e.  B )
103102adantr 465 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  .0.  e.  B )
104 ifcl 3981 . . . . . 6  |-  ( ( ( F `  k
)  e.  B  /\  .0.  e.  B )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  e.  B )
10575, 103, 104syl2anc 661 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  e.  B )
106 ifcl 3981 . . . . . 6  |-  ( ( ( F `  k
)  e.  B  /\  .0.  e.  B )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  e.  B )
10775, 103, 106syl2anc 661 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  e.  B )
108 eqidd 2468 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) )  =  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) )
109 eqidd 2468 . . . . 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 6541 . . . 4  |-  ( ph  ->  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  oF  .+  (
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 2511 . . 3  |-  ( ph  ->  F  =  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  oF  .+  ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) )
112111oveq2d 6301 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  oF  .+  (
k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) ) )
11358reseq1d 5272 . . . . . 6  |-  ( ph  ->  ( F  |`  C )  =  ( ( k  e.  A  |->  ( F `
 k ) )  |`  C ) )
114 ssun1 3667 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
115114, 93syl5sseqr 3553 . . . . . . 7  |-  ( ph  ->  C  C_  A )
11659mpteq2ia 4529 . . . . . . . 8  |-  ( k  e.  C  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  C  |->  ( F `
 k ) )
117 resmpt 5323 . . . . . . . 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 5323 . . . . . . . 8  |-  ( C 
C_  A  ->  (
( k  e.  A  |->  ( F `  k
) )  |`  C )  =  ( k  e.  C  |->  ( F `  k ) ) )
119116, 117, 1183eqtr4a 2534 . . . . . . 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 2511 . . . . 5  |-  ( ph  ->  ( F  |`  C )  =  ( ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  |`  C ) )
122121oveq2d 6301 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  =  ( G 
gsumg  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  |`  C ) ) )
123105, 51fmptd 6046 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) : A --> B )
124 frn 5737 . . . . . . 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 16189 . . . . . 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 661 . . . . 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 3627 . . . . . . . 8  |-  ( k  e.  ( A  \  C )  ->  -.  k  e.  C )
129128adantl 466 . . . . . . 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.  )
131130suppss2OLD 6515 . . . . 5  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  C
)
1321, 2, 4, 5, 6, 123, 127, 131, 17gsumzresOLD 16733 . . . 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 2508 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  =  ( G 
gsumg  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) ) )
13458reseq1d 5272 . . . . . 6  |-  ( ph  ->  ( F  |`  D )  =  ( ( k  e.  A  |->  ( F `
 k ) )  |`  D ) )
135 ssun2 3668 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
136135, 93syl5sseqr 3553 . . . . . . 7  |-  ( ph  ->  D  C_  A )
13785mpteq2ia 4529 . . . . . . . 8  |-  ( k  e.  D  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  D  |->  ( F `
 k ) )
138 resmpt 5323 . . . . . . . 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 5323 . . . . . . . 8  |-  ( D 
C_  A  ->  (
( k  e.  A  |->  ( F `  k
) )  |`  D )  =  ( k  e.  D  |->  ( F `  k ) ) )
140137, 138, 1393eqtr4a 2534 . . . . . . 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 2511 . . . . 5  |-  ( ph  ->  ( F  |`  D )  =  ( ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  |`  D ) )
143142oveq2d 6301 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  D ) )  =  ( G 
gsumg  ( ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  |`  D ) ) )
144107, 55fmptd 6046 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) : A --> B )
145 frn 5737 . . . . . . 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 16189 . . . . . 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 661 . . . . 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 3627 . . . . . . . 8  |-  ( k  e.  ( A  \  D )  ->  -.  k  e.  D )
150149adantl 466 . . . . . . 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.  )
152151suppss2OLD 6515 . . . . 5  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  D
)
1531, 2, 4, 5, 6, 144, 148, 152, 23gsumzresOLD 16733 . . . 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 2508 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  D ) )  =  ( G 
gsumg  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) ) )
155133, 154oveq12d 6303 . 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 2518 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( ( G  gsumg  ( F  |`  C ) )  .+  ( G 
gsumg  ( F  |`  D ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   ifcif 3939   {csn 4027    |-> cmpt 4505   `'ccnv 4998   ran crn 5000    |` cres 5001   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6285    oFcof 6523   Fincfn 7517   Basecbs 14493   ↾s cress 14494   +g cplusg 14558   0gc0g 14698    gsumg cgsu 14699  Moorecmre 14840  mrClscmrc 14841  ACScacs 14843   Mndcmnd 15729  SubMndcsubmnd 15788  Cntzccntz 16167  CMndccmn 16613
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 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-seq 12077  df-hash 12375  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-0g 14700  df-gsum 14701  df-mre 14844  df-mrc 14845  df-acs 14847  df-mnd 15735  df-submnd 15790  df-cntz 16169  df-cmn 16615
This theorem is referenced by:  gsumsplitOLD  16762  dpjidclOLD  16928
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