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Theorem gsumzsplit 16534
Description: Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)
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 finSupp  .0.  )
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.f . . . 4  |-  ( ph  ->  F : A --> B )
8 fvex 5804 . . . . . 6  |-  ( 0g
`  G )  e. 
_V
92, 8eqeltri 2536 . . . . 5  |-  .0.  e.  _V
109a1i 11 . . . 4  |-  ( ph  ->  .0.  e.  _V )
11 gsumzsplit.w . . . 4  |-  ( ph  ->  F finSupp  .0.  )
127, 6, 10, 11fsuppmptif 7755 . . 3  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) finSupp  .0.  )
137, 6, 10, 11fsuppmptif 7755 . . 3  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) finSupp  .0.  )
141submacs 15607 . . . . 5  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
15 acsmre 14704 . . . . 5  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
165, 14, 153syl 20 . . . 4  |-  ( ph  ->  (SubMnd `  G )  e.  (Moore `  B )
)
17 frn 5668 . . . . 5  |-  ( F : A --> B  ->  ran  F  C_  B )
187, 17syl 16 . . . 4  |-  ( ph  ->  ran  F  C_  B
)
19 eqid 2452 . . . . 5  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
2019mrccl 14663 . . . 4  |-  ( ( (SubMnd `  G )  e.  (Moore `  B )  /\  ran  F  C_  B
)  ->  ( (mrCls `  (SubMnd `  G )
) `  ran  F )  e.  (SubMnd `  G
) )
2116, 18, 20syl2anc 661 . . 3  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  e.  (SubMnd `  G ) )
22 gsumzsplit.c . . . . 5  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
23 eqid 2452 . . . . . 6  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
244, 19, 23cntzspan 16442 . . . . 5  |-  ( ( G  e.  Mnd  /\  ran  F  C_  ( Z `  ran  F ) )  ->  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  e. CMnd )
255, 22, 24syl2anc 661 . . . 4  |-  ( ph  ->  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)  e. CMnd )
2623, 4submcmn2 16439 . . . . 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 ) ) ) )
2721, 26syl 16 . . . 4  |-  ( ph  ->  ( ( Gs  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  e. CMnd  <->  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) 
C_  ( Z `  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) ) ) )
2825, 27mpbid 210 . . 3  |-  ( ph  ->  ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  C_  ( Z `  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) ) )
2916, 19, 18mrcssidd 14677 . . . . . . 7  |-  ( ph  ->  ran  F  C_  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) )
3029adantr 465 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ran  F 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
31 ffn 5662 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
327, 31syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  A )
33 fnfvelrn 5944 . . . . . . 7  |-  ( ( F  Fn  A  /\  k  e.  A )  ->  ( F `  k
)  e.  ran  F
)
3432, 33sylan 471 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  ran  F )
3530, 34sseldd 3460 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
362subm0cl 15594 . . . . . . 7  |-  ( ( (mrCls `  (SubMnd `  G
) ) `  ran  F )  e.  (SubMnd `  G )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
3721, 36syl 16 . . . . . 6  |-  ( ph  ->  .0.  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
3837adantr 465 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  .0.  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
3935, 38ifcld 3935 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
40 eqid 2452 . . . 4  |-  ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)
4139, 40fmptd 5971 . . 3  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) : A --> ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
4235, 38ifcld 3935 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )
43 eqid 2452 . . . 4  |-  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)
4442, 43fmptd 5971 . . 3  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) : A --> ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
451, 2, 3, 4, 5, 6, 12, 13, 21, 28, 41, 44gsumzadd 16525 . 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.  )
) ) ) )
467feqmptd 5848 . . . . 5  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( F `
 k ) ) )
47 iftrue 3900 . . . . . . . . . 10  |-  ( k  e.  C  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
4847adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
49 gsumzsplit.i . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
50 noel 3744 . . . . . . . . . . . . . . . 16  |-  -.  k  e.  (/)
51 eleq2 2525 . . . . . . . . . . . . . . . 16  |-  ( ( C  i^i  D )  =  (/)  ->  ( k  e.  ( C  i^i  D )  <->  k  e.  (/) ) )
5250, 51mtbiri 303 . . . . . . . . . . . . . . 15  |-  ( ( C  i^i  D )  =  (/)  ->  -.  k  e.  ( C  i^i  D
) )
5349, 52syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  k  e.  ( C  i^i  D ) )
5453adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  A )  ->  -.  k  e.  ( C  i^i  D ) )
55 elin 3642 . . . . . . . . . . . . 13  |-  ( k  e.  ( C  i^i  D )  <->  ( k  e.  C  /\  k  e.  D ) )
5654, 55sylnib 304 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  A )  ->  -.  ( k  e.  C  /\  k  e.  D
) )
57 imnan 422 . . . . . . . . . . . 12  |-  ( ( k  e.  C  ->  -.  k  e.  D
)  <->  -.  ( k  e.  C  /\  k  e.  D ) )
5856, 57sylibr 212 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  (
k  e.  C  ->  -.  k  e.  D
) )
5958imp 429 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  -.  k  e.  D )
60 iffalse 3902 . . . . . . . . . 10  |-  ( -.  k  e.  D  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  .0.  )
6159, 60syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  .0.  )
6248, 61oveq12d 6213 . . . . . . . 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.  ) )
637ffvelrnda 5947 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  e.  B )
641, 3, 2mndrid 15556 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  ( F `  k )  e.  B )  -> 
( ( F `  k )  .+  .0.  )  =  ( F `  k ) )
655, 64sylan 471 . . . . . . . . . 10  |-  ( (
ph  /\  ( F `  k )  e.  B
)  ->  ( ( F `  k )  .+  .0.  )  =  ( F `  k ) )
6663, 65syldan 470 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  (
( F `  k
)  .+  .0.  )  =  ( F `  k ) )
6766adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  (
( F `  k
)  .+  .0.  )  =  ( F `  k ) )
6862, 67eqtrd 2493 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  C )  ->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  =  ( F `  k ) )
6958con2d 115 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  (
k  e.  D  ->  -.  k  e.  C
) )
7069imp 429 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  -.  k  e.  C )
71 iffalse 3902 . . . . . . . . . 10  |-  ( -.  k  e.  C  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  .0.  )
7270, 71syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  =  .0.  )
73 iftrue 3900 . . . . . . . . . 10  |-  ( k  e.  D  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
7473adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  =  ( F `  k ) )
7572, 74oveq12d 6213 . . . . . . . 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 ) ) )
761, 3, 2mndlid 15555 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  ( F `  k )  e.  B )  -> 
(  .0.  .+  ( F `  k )
)  =  ( F `
 k ) )
775, 76sylan 471 . . . . . . . . . 10  |-  ( (
ph  /\  ( F `  k )  e.  B
)  ->  (  .0.  .+  ( F `  k
) )  =  ( F `  k ) )
7863, 77syldan 470 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  (  .0.  .+  ( F `  k ) )  =  ( F `  k
) )
7978adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  (  .0.  .+  ( F `  k ) )  =  ( F `  k
) )
8075, 79eqtrd 2493 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  A )  /\  k  e.  D )  ->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  =  ( F `  k ) )
81 gsumzsplit.u . . . . . . . . . 10  |-  ( ph  ->  A  =  ( C  u.  D ) )
8281eleq2d 2522 . . . . . . . . 9  |-  ( ph  ->  ( k  e.  A  <->  k  e.  ( C  u.  D ) ) )
83 elun 3600 . . . . . . . . 9  |-  ( k  e.  ( C  u.  D )  <->  ( k  e.  C  \/  k  e.  D ) )
8482, 83syl6bb 261 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  <->  ( k  e.  C  \/  k  e.  D )
) )
8584biimpa 484 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  (
k  e.  C  \/  k  e.  D )
)
8668, 80, 85mpjaodan 784 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  =  ( F `  k ) )
8786mpteq2dva 4481 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) )  =  ( k  e.  A  |->  ( F `  k
) ) )
8846, 87eqtr4d 2496 . . . 4  |-  ( ph  ->  F  =  ( k  e.  A  |->  ( if ( k  e.  C ,  ( F `  k ) ,  .0.  )  .+  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) )
891, 2mndidcl 15553 . . . . . . . 8  |-  ( G  e.  Mnd  ->  .0.  e.  B )
905, 89syl 16 . . . . . . 7  |-  ( ph  ->  .0.  e.  B )
9190adantr 465 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  .0.  e.  B )
9263, 91ifcld 3935 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  C ,  ( F `  k ) ,  .0.  )  e.  B )
9363, 91ifcld 3935 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  if ( k  e.  D ,  ( F `  k ) ,  .0.  )  e.  B )
94 eqidd 2453 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) )  =  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) )
95 eqidd 2453 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) )  =  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) )
966, 92, 93, 94, 95offval2 6441 . . . 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.  ) ) ) )
9788, 96eqtr4d 2496 . . 3  |-  ( ph  ->  F  =  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  oF  .+  ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) ) )
9897oveq2d 6211 . 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.  ) ) ) ) )
9946reseq1d 5212 . . . . . 6  |-  ( ph  ->  ( F  |`  C )  =  ( ( k  e.  A  |->  ( F `
 k ) )  |`  C ) )
100 ssun1 3622 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
101100, 81syl5sseqr 3508 . . . . . . 7  |-  ( ph  ->  C  C_  A )
10247mpteq2ia 4477 . . . . . . . 8  |-  ( k  e.  C  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  C  |->  ( F `
 k ) )
103 resmpt 5259 . . . . . . . 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.  ) ) )
104 resmpt 5259 . . . . . . . 8  |-  ( C 
C_  A  ->  (
( k  e.  A  |->  ( F `  k
) )  |`  C )  =  ( k  e.  C  |->  ( F `  k ) ) )
105102, 103, 1043eqtr4a 2519 . . . . . . 7  |-  ( C 
C_  A  ->  (
( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) )  |`  C )  =  ( ( k  e.  A  |->  ( F `  k
) )  |`  C ) )
106101, 105syl 16 . . . . . 6  |-  ( ph  ->  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  |`  C )  =  ( ( k  e.  A  |->  ( F `  k
) )  |`  C ) )
10799, 106eqtr4d 2496 . . . . 5  |-  ( ph  ->  ( F  |`  C )  =  ( ( k  e.  A  |->  if ( k  e.  C , 
( F `  k
) ,  .0.  )
)  |`  C ) )
108107oveq2d 6211 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  =  ( G 
gsumg  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) )  |`  C ) ) )
10992, 40fmptd 5971 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) : A --> B )
110 frn 5668 . . . . . . 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 ) )
11141, 110syl 16 . . . . . 6  |-  ( ph  ->  ran  ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) ) 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
1124cntzidss 15969 . . . . . 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.  ) ) ) )
11328, 111, 112syl2anc 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.  ) ) ) )
114 eldifn 3582 . . . . . . . 8  |-  ( k  e.  ( A  \  C )  ->  -.  k  e.  C )
115114adantl 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  C ) )  ->  -.  k  e.  C )
116115, 71syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  C ) )  ->  if (
k  e.  C , 
( F `  k
) ,  .0.  )  =  .0.  )
117116, 6suppss2 6828 . . . . 5  |-  ( ph  ->  ( ( k  e.  A  |->  if ( k  e.  C ,  ( F `  k ) ,  .0.  ) ) supp 
.0.  )  C_  C
)
1181, 2, 4, 5, 6, 109, 113, 117, 12gsumzres 16504 . . . 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.  ) ) ) )
119108, 118eqtrd 2493 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  C ) )  =  ( G 
gsumg  ( k  e.  A  |->  if ( k  e.  C ,  ( F `
 k ) ,  .0.  ) ) ) )
12046reseq1d 5212 . . . . . 6  |-  ( ph  ->  ( F  |`  D )  =  ( ( k  e.  A  |->  ( F `
 k ) )  |`  D ) )
121 ssun2 3623 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
122121, 81syl5sseqr 3508 . . . . . . 7  |-  ( ph  ->  D  C_  A )
12373mpteq2ia 4477 . . . . . . . 8  |-  ( k  e.  D  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  =  ( k  e.  D  |->  ( F `
 k ) )
124 resmpt 5259 . . . . . . . 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.  ) ) )
125 resmpt 5259 . . . . . . . 8  |-  ( D 
C_  A  ->  (
( k  e.  A  |->  ( F `  k
) )  |`  D )  =  ( k  e.  D  |->  ( F `  k ) ) )
126123, 124, 1253eqtr4a 2519 . . . . . . 7  |-  ( D 
C_  A  ->  (
( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) )  |`  D )  =  ( ( k  e.  A  |->  ( F `  k
) )  |`  D ) )
127122, 126syl 16 . . . . . 6  |-  ( ph  ->  ( ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  |`  D )  =  ( ( k  e.  A  |->  ( F `  k
) )  |`  D ) )
128120, 127eqtr4d 2496 . . . . 5  |-  ( ph  ->  ( F  |`  D )  =  ( ( k  e.  A  |->  if ( k  e.  D , 
( F `  k
) ,  .0.  )
)  |`  D ) )
129128oveq2d 6211 . . . 4  |-  ( ph  ->  ( G  gsumg  ( F  |`  D ) )  =  ( G 
gsumg  ( ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) )  |`  D ) ) )
13093, 43fmptd 5971 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) : A --> B )
131 frn 5668 . . . . . . 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 ) )
13244, 131syl 16 . . . . . 6  |-  ( ph  ->  ran  ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) 
C_  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
1334cntzidss 15969 . . . . . 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.  ) ) ) )
13428, 132, 133syl2anc 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.  ) ) ) )
135 eldifn 3582 . . . . . . . 8  |-  ( k  e.  ( A  \  D )  ->  -.  k  e.  D )
136135adantl 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( A  \  D ) )  ->  -.  k  e.  D )
137136, 60syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( A  \  D ) )  ->  if (
k  e.  D , 
( F `  k
) ,  .0.  )  =  .0.  )
138137, 6suppss2 6828 . . . . 5  |-  ( ph  ->  ( ( k  e.  A  |->  if ( k  e.  D ,  ( F `  k ) ,  .0.  ) ) supp 
.0.  )  C_  D
)
1391, 2, 4, 5, 6, 130, 134, 138, 13gsumzres 16504 . . . 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.  ) ) ) )
140129, 139eqtrd 2493 . . 3  |-  ( ph  ->  ( G  gsumg  ( F  |`  D ) )  =  ( G 
gsumg  ( k  e.  A  |->  if ( k  e.  D ,  ( F `
 k ) ,  .0.  ) ) ) )
141119, 140oveq12d 6213 . 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.  )
) ) ) )
14245, 98, 1413eqtr4d 2503 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 3072    \ cdif 3428    u. cun 3429    i^i cin 3430    C_ wss 3431   (/)c0 3740   ifcif 3894   class class class wbr 4395    |-> cmpt 4453   ran crn 4944    |` cres 4945    Fn wfn 5516   -->wf 5517   ` cfv 5521  (class class class)co 6195    oFcof 6423   finSupp cfsupp 7726   Basecbs 14287   ↾s cress 14288   +g cplusg 14352   0gc0g 14492    gsumg cgsu 14493  Moorecmre 14634  mrClscmrc 14635  ACScacs 14637   Mndcmnd 15523  SubMndcsubmnd 15577  Cntzccntz 15947  CMndccmn 16393
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-supp 6796  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fsupp 7727  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-fzo 11661  df-seq 11919  df-hash 12216  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-0g 14494  df-gsum 14495  df-mre 14638  df-mrc 14639  df-acs 14641  df-mnd 15529  df-submnd 15579  df-cntz 15949  df-cmn 16395
This theorem is referenced by:  gsumsplit  16536  gsumzunsnd  16568  dpjidcl  16674
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