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Theorem gsumzsplitOLD 16541
Description: Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of gsumzsplit 16540 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 3484 . . . . . . . . 9  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
109a1i 11 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
118, 10suppssrOLD 5947 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F `  k )  =  .0.  )
1211ifeq1d 3916 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  if ( k  e.  C ,  .0.  ,  .0.  ) )
13 ifid 3935 . . . . . 6  |-  if ( k  e.  C ,  .0.  ,  .0.  )  =  .0.
1412, 13syl6eq 2511 . . . . 5  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  .0.  )
1514suppss2OLD 6426 . . . 4  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
16 ssfi 7645 . . . 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 3916 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  if ( k  e.  D ,  .0.  ,  .0.  ) )
19 ifid 3935 . . . . . 6  |-  if ( k  e.  D ,  .0.  ,  .0.  )  =  .0.
2018, 19syl6eq 2511 . . . . 5  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  .0.  )
2120suppss2OLD 6426 . . . 4  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
22 ssfi 7645 . . . 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 15613 . . . . 5  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
25 acsmre 14710 . . . . 5  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
265, 24, 253syl 20 . . . 4  |-  ( ph  ->  (SubMnd `  G )  e.  (Moore `  B )
)
27 frn 5674 . . . . 5  |-  ( F : A --> B  ->  ran  F  C_  B )
288, 27syl 16 . . . 4  |-  ( ph  ->  ran  F  C_  B
)
29 eqid 2454 . . . . 5  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
3029mrccl 14669 . . . 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 2454 . . . . . 6  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
344, 29, 33cntzspan 16448 . . . . 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 16445 . . . . 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 14683 . . . . . . 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 5668 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
428, 41syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  A )
43 fnfvelrn 5950 . . . . . . 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 3466 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
462subm0cl 15600 . . . . . . 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 3940 . . . . 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 2454 . . . 4  |-  ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)
5250, 51fmptd 5977 . . 3  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) : A --> ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
53 ifcl 3940 . . . . 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 2454 . . . 4  |-  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)
5654, 55fmptd 5977 . . 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 16533 . 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 5854 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
59 iftrue 3906 . . . . . . . . . 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 3750 . . . . . . . . . . . . . . . 16  |-  -.  k  e.  (/)
63 eleq2 2527 . . . . . . . . . . . . . . . 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 3648 . . . . . . . . . . . . 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 3908 . . . . . . . . . 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 6219 . . . . . . . 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 5953 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  B )
761, 3, 2mndrid 15562 . . . . . . . . . . 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 2495 . . . . . . 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 3908 . . . . . . . . . 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 3906 . . . . . . . . . 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 6219 . . . . . . . 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 15561 . . . . . . . . . . 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 2495 . . . . . . 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 2524 . . . . . . . . 9  |-  ( ph  ->  ( k  e.  A  <->  k  e.  ( C  u.  D ) ) )
95 elun 3606 . . . . . . . . 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 4487 . . . . 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 2498 . . . 4  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) )
1011, 2mndidcl 15559 . . . . . . . 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 3940 . . . . . 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 3940 . . . . . 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 2455 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) )  =  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) )
109 eqidd 2455 . . . . 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 6447 . . . 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 2498 . . 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 6217 . 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 5218 . . . . . 6  |-  ( ph  ->  ( F  |`  C )  =  ( ( k  e.  A  |->  ( F `
 k ) )  |`  C ) )
114 ssun1 3628 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
115114, 93syl5sseqr 3514 . . . . . . 7  |-  ( ph  ->  C  C_  A )
11659mpteq2ia 4483 . . . . . . . 8  |-  ( k  e.  C  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  C  |->  ( F `
 k ) )
117 resmpt 5265 . . . . . . . 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 5265 . . . . . . . 8  |-  ( C 
C_  A  ->  (
( k  e.  A  |->  ( F `  k
) )  |`  C )  =  ( k  e.  C  |->  ( F `  k ) ) )
119116, 117, 1183eqtr4a 2521 . . . . . . 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 2498 . . . . 5  |-  ( ph  ->  ( F  |`  C )  =  ( ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  |`  C ) )
122121oveq2d 6217 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  =  ( G 
gsumg  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  |`  C ) ) )
123105, 51fmptd 5977 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) : A --> B )
124 frn 5674 . . . . . . 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 15975 . . . . . 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 3588 . . . . . . . 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 6426 . . . . 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 16514 . . . 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 2495 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  =  ( G 
gsumg  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) ) )
13458reseq1d 5218 . . . . . 6  |-  ( ph  ->  ( F  |`  D )  =  ( ( k  e.  A  |->  ( F `
 k ) )  |`  D ) )
135 ssun2 3629 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
136135, 93syl5sseqr 3514 . . . . . . 7  |-  ( ph  ->  D  C_  A )
13785mpteq2ia 4483 . . . . . . . 8  |-  ( k  e.  D  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  D  |->  ( F `
 k ) )
138 resmpt 5265 . . . . . . . 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 5265 . . . . . . . 8  |-  ( D 
C_  A  ->  (
( k  e.  A  |->  ( F `  k
) )  |`  D )  =  ( k  e.  D  |->  ( F `  k ) ) )
140137, 138, 1393eqtr4a 2521 . . . . . . 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 2498 . . . . 5  |-  ( ph  ->  ( F  |`  D )  =  ( ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  |`  D ) )
143142oveq2d 6217 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  D ) )  =  ( G 
gsumg  ( ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  |`  D ) ) )
144107, 55fmptd 5977 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) : A --> B )
145 frn 5674 . . . . . . 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 15975 . . . . . 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 3588 . . . . . . . 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 6426 . . . . 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 16514 . . . 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 2495 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  D ) )  =  ( G 
gsumg  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) ) )
155133, 154oveq12d 6219 . 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 2505 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 1370    e. wcel 1758   _Vcvv 3078    \ cdif 3434    u. cun 3435    i^i cin 3436    C_ wss 3437   (/)c0 3746   ifcif 3900   {csn 3986    |-> cmpt 4459   `'ccnv 4948   ran crn 4950    |` cres 4951   "cima 4952    Fn wfn 5522   -->wf 5523   ` cfv 5527  (class class class)co 6201    oFcof 6429   Fincfn 7421   Basecbs 14293   ↾s cress 14294   +g cplusg 14358   0gc0g 14498    gsumg cgsu 14499  Moorecmre 14640  mrClscmrc 14641  ACScacs 14643   Mndcmnd 15529  SubMndcsubmnd 15583  Cntzccntz 15953  CMndccmn 16399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-seq 11925  df-hash 12222  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-0g 14500  df-gsum 14501  df-mre 14644  df-mrc 14645  df-acs 14647  df-mnd 15535  df-submnd 15585  df-cntz 15955  df-cmn 16401
This theorem is referenced by:  gsumsplitOLD  16543  dpjidclOLD  16687
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