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Theorem gsumzsplit 15208
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 3199 . . . . . . . . 9  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
109a1i 10 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
118, 10suppssr 5661 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F `  k )  =  .0.  )
1211ifeq1d 3581 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  if ( k  e.  C ,  .0.  ,  .0.  ) )
13 ifid 3599 . . . . . 6  |-  if ( k  e.  C ,  .0.  ,  .0.  )  =  .0.
1412, 13syl6eq 2333 . . . . 5  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  .0.  )
1514suppss2 6075 . . . 4  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
16 ssfi 7085 . . . 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 642 . . 3  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
1811ifeq1d 3581 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  if ( k  e.  D ,  .0.  ,  .0.  ) )
19 ifid 3599 . . . . . 6  |-  if ( k  e.  D ,  .0.  ,  .0.  )  =  .0.
2018, 19syl6eq 2333 . . . . 5  |-  ( (
ph  /\  k  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  .0.  )
2120suppss2 6075 . . . 4  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
22 ssfi 7085 . . . 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 642 . . 3  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
241submacs 14444 . . . . 5  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
25 acsmre 13556 . . . . 5  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
265, 24, 253syl 18 . . . 4  |-  ( ph  ->  (SubMnd `  G )  e.  (Moore `  B )
)
27 frn 5397 . . . . 5  |-  ( F : A --> B  ->  ran  F  C_  B )
288, 27syl 15 . . . 4  |-  ( ph  ->  ran  F  C_  B
)
29 eqid 2285 . . . . 5  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
3029mrccl 13515 . . . 4  |-  ( ( (SubMnd `  G )  e.  (Moore `  B )  /\  ran  F  C_  B
)  ->  ( (mrCls `  (SubMnd `  G )
) `  ran  F )  e.  (SubMnd `  G
) )
3126, 28, 30syl2anc 642 . . 3  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  e.  (SubMnd `  G ) )
32 gsumzsplit.c . . . . 5  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
33 eqid 2285 . . . . . 6  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
344, 29, 33cntzspan 15139 . . . . 5  |-  ( ( G  e.  Mnd  /\  ran  F  C_  ( Z `  ran  F ) )  ->  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  e. CMnd )
355, 32, 34syl2anc 642 . . . 4  |-  ( ph  ->  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)  e. CMnd )
3633, 4submcmn2 15137 . . . . 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 15 . . . 4  |-  ( ph  ->  ( ( Gs  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  e. CMnd  <->  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) 
C_  ( Z `  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) ) ) )
3835, 37mpbid 201 . . 3  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  C_  ( Z `  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) ) )
3929mrcssid 13521 . . . . . . . 8  |-  ( ( (SubMnd `  G )  e.  (Moore `  B )  /\  ran  F  C_  B
)  ->  ran  F  C_  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )
4026, 28, 39syl2anc 642 . . . . . . 7  |-  ( ph  ->  ran  F  C_  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) )
4140adantr 451 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ran  F 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
42 ffn 5391 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
438, 42syl 15 . . . . . . 7  |-  ( ph  ->  F  Fn  A )
44 fnfvelrn 5664 . . . . . . 7  |-  ( ( F  Fn  A  /\  k  e.  A )  ->  ( F `  k
)  e.  ran  F
)
4543, 44sylan 457 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  ran  F )
4641, 45sseldd 3183 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
472subm0cl 14431 . . . . . . 7  |-  ( ( (mrCls `  (SubMnd `  G
) ) `  ran  F )  e.  (SubMnd `  G )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
4831, 47syl 15 . . . . . 6  |-  ( ph  ->  .0.  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
4948adantr 451 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
50 ifcl 3603 . . . . 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 ) )
5146, 49, 50syl2anc 642 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
52 eqid 2285 . . . 4  |-  ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)
5351, 52fmptd 5686 . . 3  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) : A --> ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
54 ifcl 3603 . . . . 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 ) )
5546, 49, 54syl2anc 642 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
56 eqid 2285 . . . 4  |-  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)
5755, 56fmptd 5686 . . 3  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) : A --> ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
581, 2, 3, 4, 5, 6, 17, 23, 31, 38, 53, 57gsumzadd 15206 . 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.  )
) ) ) )
598feqmptd 5577 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
60 iftrue 3573 . . . . . . . . . 10  |-  ( k  e.  C  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
6160adantl 452 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
62 gsumzsplit.i . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
63 noel 3461 . . . . . . . . . . . . . . . 16  |-  -.  k  e.  (/)
64 eleq2 2346 . . . . . . . . . . . . . . . 16  |-  ( ( C  i^i  D )  =  (/)  ->  ( k  e.  ( C  i^i  D )  <->  k  e.  (/) ) )
6563, 64mtbiri 294 . . . . . . . . . . . . . . 15  |-  ( ( C  i^i  D )  =  (/)  ->  -.  k  e.  ( C  i^i  D
) )
6662, 65syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  k  e.  ( C  i^i  D ) )
6766adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  A )  ->  -.  k  e.  ( C  i^i  D ) )
68 elin 3360 . . . . . . . . . . . . 13  |-  ( k  e.  ( C  i^i  D )  <->  ( k  e.  C  /\  k  e.  D ) )
6967, 68sylnib 295 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  A )  ->  -.  ( k  e.  C  /\  k  e.  D
) )
70 imnan 411 . . . . . . . . . . . 12  |-  ( ( k  e.  C  ->  -.  k  e.  D
)  <->  -.  ( k  e.  C  /\  k  e.  D ) )
7169, 70sylibr 203 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  (
k  e.  C  ->  -.  k  e.  D
) )
7271imp 418 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  -.  k  e.  D )
73 iffalse 3574 . . . . . . . . . 10  |-  ( -.  k  e.  D  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  .0.  )
7472, 73syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  .0.  )
7561, 74oveq12d 5878 . . . . . . . 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.  ) )
76 ffvelrn 5665 . . . . . . . . . . 11  |-  ( ( F : A --> B  /\  k  e.  A )  ->  ( F `  k
)  e.  B )
778, 76sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  B )
781, 3, 2mndrid 14396 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  ( F `  k )  e.  B )  -> 
( ( F `  k )  .+  .0.  )  =  ( F `  k ) )
795, 78sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  ( F `  k )  e.  B
)  ->  ( ( F `  k )  .+  .0.  )  =  ( F `  k ) )
8077, 79syldan 456 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  .+  .0.  )  =  ( F `  k ) )
8180adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  (
( F `  k
)  .+  .0.  )  =  ( F `  k ) )
8275, 81eqtrd 2317 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  =  ( F `  k ) )
8371con2d 107 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  (
k  e.  D  ->  -.  k  e.  C
) )
8483imp 418 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  -.  k  e.  C )
85 iffalse 3574 . . . . . . . . . 10  |-  ( -.  k  e.  C  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  .0.  )
8684, 85syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  .0.  )
87 iftrue 3573 . . . . . . . . . 10  |-  ( k  e.  D  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
8887adantl 452 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
8986, 88oveq12d 5878 . . . . . . . 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 ) ) )
901, 3, 2mndlid 14395 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  ( F `  k )  e.  B )  -> 
(  .0.  .+  ( F `  k )
)  =  ( F `
 k ) )
915, 90sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  ( F `  k )  e.  B
)  ->  (  .0.  .+  ( F `  k
) )  =  ( F `  k ) )
9277, 91syldan 456 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  (  .0.  .+  ( F `  k ) )  =  ( F `  k
) )
9392adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  (  .0.  .+  ( F `  k ) )  =  ( F `  k
) )
9489, 93eqtrd 2317 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  =  ( F `  k ) )
95 gsumzsplit.u . . . . . . . . . 10  |-  ( ph  ->  A  =  ( C  u.  D ) )
9695eleq2d 2352 . . . . . . . . 9  |-  ( ph  ->  ( k  e.  A  <->  k  e.  ( C  u.  D ) ) )
97 elun 3318 . . . . . . . . 9  |-  ( k  e.  ( C  u.  D )  <->  ( k  e.  C  \/  k  e.  D ) )
9896, 97syl6bb 252 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  <->  ( k  e.  C  \/  k  e.  D )
) )
9998biimpa 470 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  (
k  e.  C  \/  k  e.  D )
)
10082, 94, 99mpjaodan 761 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  =  ( F `  k ) )
101100mpteq2dva 4108 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) )  =  ( k  e.  A  |->  ( F `  k
) ) )
10259, 101eqtr4d 2320 . . . 4  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) )
1031, 2mndidcl 14393 . . . . . . . 8  |-  ( G  e.  Mnd  ->  .0.  e.  B )
1045, 103syl 15 . . . . . . 7  |-  ( ph  ->  .0.  e.  B )
105104adantr 451 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  .0.  e.  B )
106 ifcl 3603 . . . . . 6  |-  ( ( ( F `  k
)  e.  B  /\  .0.  e.  B )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  e.  B )
10777, 105, 106syl2anc 642 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  e.  B )
108 ifcl 3603 . . . . . 6  |-  ( ( ( F `  k
)  e.  B  /\  .0.  e.  B )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  e.  B )
10977, 105, 108syl2anc 642 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  e.  B )
110 eqidd 2286 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) )  =  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) )
111 eqidd 2286 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) )  =  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) )
1126, 107, 109, 110, 111offval2 6097 . . . 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.  ) ) ) )
113102, 112eqtr4d 2320 . . 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.  ) ) ) )
114113oveq2d 5876 . 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.  ) ) ) ) )
11559reseq1d 4956 . . . . . 6  |-  ( ph  ->  ( F  |`  C )  =  ( ( k  e.  A  |->  ( F `
 k ) )  |`  C ) )
116 ssun1 3340 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
117116, 95syl5sseqr 3229 . . . . . . 7  |-  ( ph  ->  C  C_  A )
11860mpteq2ia 4104 . . . . . . . 8  |-  ( k  e.  C  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  C  |->  ( F `
 k ) )
119 resmpt 5002 . . . . . . . 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.  ) ) )
120 resmpt 5002 . . . . . . . 8  |-  ( C 
C_  A  ->  (
( k  e.  A  |->  ( F `  k
) )  |`  C )  =  ( k  e.  C  |->  ( F `  k ) ) )
121118, 119, 1203eqtr4a 2343 . . . . . . 7  |-  ( C 
C_  A  ->  (
( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) )  |`  C )  =  ( ( k  e.  A  |->  ( F `  k
) )  |`  C ) )
122117, 121syl 15 . . . . . 6  |-  ( ph  ->  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  |`  C )  =  ( ( k  e.  A  |->  ( F `  k
) )  |`  C ) )
123115, 122eqtr4d 2320 . . . . 5  |-  ( ph  ->  ( F  |`  C )  =  ( ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  |`  C ) )
124123oveq2d 5876 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  =  ( G 
gsumg  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  |`  C ) ) )
125107, 52fmptd 5686 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) : A --> B )
126 frn 5397 . . . . . . 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 ) )
12753, 126syl 15 . . . . . 6  |-  ( ph  ->  ran  ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) ) 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
1284cntzidss 14815 . . . . . 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.  ) ) ) )
12938, 127, 128syl2anc 642 . . . . 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.  ) ) ) )
130 eldifn 3301 . . . . . . . 8  |-  ( k  e.  ( A  \  C )  ->  -.  k  e.  C )
131130adantl 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  C ) )  ->  -.  k  e.  C )
132131, 85syl 15 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  C ) )  ->  if (
k  e.  C , 
( F `  k
) ,  .0.  )  =  .0.  )
133132suppss2 6075 . . . . 5  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  C
)
1341, 2, 4, 5, 6, 125, 129, 133, 17gsumzres 15196 . . . 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.  ) ) ) )
135124, 134eqtrd 2317 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  =  ( G 
gsumg  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) ) )
13659reseq1d 4956 . . . . . 6  |-  ( ph  ->  ( F  |`  D )  =  ( ( k  e.  A  |->  ( F `
 k ) )  |`  D ) )
137 ssun2 3341 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
138137, 95syl5sseqr 3229 . . . . . . 7  |-  ( ph  ->  D  C_  A )
13987mpteq2ia 4104 . . . . . . . 8  |-  ( k  e.  D  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  D  |->  ( F `
 k ) )
140 resmpt 5002 . . . . . . . 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.  ) ) )
141 resmpt 5002 . . . . . . . 8  |-  ( D 
C_  A  ->  (
( k  e.  A  |->  ( F `  k
) )  |`  D )  =  ( k  e.  D  |->  ( F `  k ) ) )
142139, 140, 1413eqtr4a 2343 . . . . . . 7  |-  ( D 
C_  A  ->  (
( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) )  |`  D )  =  ( ( k  e.  A  |->  ( F `  k
) )  |`  D ) )
143138, 142syl 15 . . . . . 6  |-  ( ph  ->  ( ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  |`  D )  =  ( ( k  e.  A  |->  ( F `  k
) )  |`  D ) )
144136, 143eqtr4d 2320 . . . . 5  |-  ( ph  ->  ( F  |`  D )  =  ( ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  |`  D ) )
145144oveq2d 5876 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  D ) )  =  ( G 
gsumg  ( ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  |`  D ) ) )
146109, 56fmptd 5686 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) : A --> B )
147 frn 5397 . . . . . . 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 ) )
14857, 147syl 15 . . . . . 6  |-  ( ph  ->  ran  ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
1494cntzidss 14815 . . . . . 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.  ) ) ) )
15038, 148, 149syl2anc 642 . . . . 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.  ) ) ) )
151 eldifn 3301 . . . . . . . 8  |-  ( k  e.  ( A  \  D )  ->  -.  k  e.  D )
152151adantl 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  D ) )  ->  -.  k  e.  D )
153152, 73syl 15 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  D ) )  ->  if (
k  e.  D , 
( F `  k
) ,  .0.  )  =  .0.  )
154153suppss2 6075 . . . . 5  |-  ( ph  ->  ( `' ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
) " ( _V 
\  {  .0.  }
) )  C_  D
)
1551, 2, 4, 5, 6, 146, 150, 154, 23gsumzres 15196 . . . 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.  ) ) ) )
156145, 155eqtrd 2317 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  D ) )  =  ( G 
gsumg  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) ) )
157135, 156oveq12d 5878 . 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.  )
) ) ) )
15858, 114, 1573eqtr4d 2327 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 176    \/ wo 357    /\ wa 358    = wceq 1625    e. wcel 1686   _Vcvv 2790    \ cdif 3151    u. cun 3152    i^i cin 3153    C_ wss 3154   (/)c0 3457   ifcif 3567   {csn 3642    e. cmpt 4079   `'ccnv 4690   ran crn 4692    |` cres 4693   "cima 4694    Fn wfn 5252   -->wf 5253   ` cfv 5257  (class class class)co 5860    o Fcof 6078   Fincfn 6865   Basecbs 13150   ↾s cress 13151   +g cplusg 13210   0gc0g 13402    gsumg cgsu 13403  Moorecmre 13486  mrClscmrc 13487  ACScacs 13489   Mndcmnd 14363  SubMndcsubmnd 14416  Cntzccntz 14793  CMndccmn 15091
This theorem is referenced by:  gsumsplit  15209  dpjidcl  15295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-oi 7227  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-fzo 10873  df-seq 11049  df-hash 11340  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-0g 13406  df-gsum 13407  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-cntz 14795  df-cmn 15093
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