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Theorem gsumzoppg 15531
Description: The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
gsumzoppg.b  |-  B  =  ( Base `  G
)
gsumzoppg.0  |-  .0.  =  ( 0g `  G )
gsumzoppg.z  |-  Z  =  (Cntz `  G )
gsumzoppg.o  |-  O  =  (oppg
`  G )
gsumzoppg.g  |-  ( ph  ->  G  e.  Mnd )
gsumzoppg.a  |-  ( ph  ->  A  e.  V )
gsumzoppg.f  |-  ( ph  ->  F : A --> B )
gsumzoppg.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzoppg.n  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
Assertion
Ref Expression
gsumzoppg  |-  ( ph  ->  ( O  gsumg  F )  =  ( G  gsumg  F ) )

Proof of Theorem gsumzoppg
Dummy variables  f 
k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzoppg.g . . . . . . . 8  |-  ( ph  ->  G  e.  Mnd )
2 gsumzoppg.o . . . . . . . . 9  |-  O  =  (oppg
`  G )
32oppgmnd 15142 . . . . . . . 8  |-  ( G  e.  Mnd  ->  O  e.  Mnd )
41, 3syl 16 . . . . . . 7  |-  ( ph  ->  O  e.  Mnd )
5 gsumzoppg.a . . . . . . 7  |-  ( ph  ->  A  e.  V )
6 gsumzoppg.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
72, 6oppgid 15144 . . . . . . . 8  |-  .0.  =  ( 0g `  O )
87gsumz 14773 . . . . . . 7  |-  ( ( O  e.  Mnd  /\  A  e.  V )  ->  ( O  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
94, 5, 8syl2anc 643 . . . . . 6  |-  ( ph  ->  ( O  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
106gsumz 14773 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
111, 5, 10syl2anc 643 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
129, 11eqtr4d 2470 . . . . 5  |-  ( ph  ->  ( O  gsumg  ( k  e.  A  |->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
1312adantr 452 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( O  gsumg  ( k  e.  A  |->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
14 gsumzoppg.f . . . . . 6  |-  ( ph  ->  F : A --> B )
15 ssid 3359 . . . . . . 7  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
1615a1i 11 . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
1714, 16gsumcllem 15508 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
1817oveq2d 6089 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( O  gsumg  F )  =  ( O  gsumg  ( k  e.  A  |->  .0.  ) ) )
1917oveq2d 6089 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
2013, 18, 193eqtr4d 2477 . . 3  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( O  gsumg  F )  =  ( G  gsumg  F ) )
2120ex 424 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  ->  ( O  gsumg  F )  =  ( G  gsumg  F ) ) )
22 simprl 733 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )
23 nnuz 10513 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2422, 23syl6eleq 2525 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  ( ZZ>= `  1 )
)
2514adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  F : A --> B )
26 ffn 5583 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  F  Fn  A )
27 dffn4 5651 . . . . . . . . . . . 12  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
2826, 27sylib 189 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F : A -onto-> ran  F
)
29 fof 5645 . . . . . . . . . . 11  |-  ( F : A -onto-> ran  F  ->  F : A --> ran  F
)
3025, 28, 293syl 19 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  F : A --> ran  F )
311adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  G  e.  Mnd )
32 gsumzoppg.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  G
)
3332submacs 14757 . . . . . . . . . . . 12  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
34 acsmre 13869 . . . . . . . . . . . 12  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
3531, 33, 343syl 19 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  (SubMnd `  G )  e.  (Moore `  B )
)
36 eqid 2435 . . . . . . . . . . 11  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
37 frn 5589 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  ran  F  C_  B )
3825, 37syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  F  C_  B
)
3935, 36, 38mrcssidd 13842 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  F  C_  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) )
40 fss 5591 . . . . . . . . . 10  |-  ( ( F : A --> ran  F  /\  ran  F  C_  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) )  ->  F : A --> ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
4130, 39, 40syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  F : A --> ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )
42 f1of1 5665 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) ) )
4342ad2antll 710 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-> ( `' F " ( _V  \  {  .0.  } ) ) )
44 cnvimass 5216 . . . . . . . . . . . 12  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
45 fdm 5587 . . . . . . . . . . . . 13  |-  ( F : A --> B  ->  dom  F  =  A )
4625, 45syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  dom  F  =  A )
4744, 46syl5sseq 3388 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  A )
48 f1ss 5636 . . . . . . . . . . 11  |-  ( ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  C_  A
)  ->  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-> A )
4943, 47, 48syl2anc 643 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-> A )
50 f1f 5631 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> A  ->  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) --> A )
5149, 50syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) --> A )
52 fco 5592 . . . . . . . . 9  |-  ( ( F : A --> ( (mrCls `  (SubMnd `  G )
) `  ran  F )  /\  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) --> A )  -> 
( F  o.  f
) : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) --> ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
5341, 51, 52syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F  o.  f ) : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) --> ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
5453ffvelrnda 5862 . . . . . . 7  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  x  e.  ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) )  ->  (
( F  o.  f
) `  x )  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
5536mrccl 13828 . . . . . . . . . 10  |-  ( ( (SubMnd `  G )  e.  (Moore `  B )  /\  ran  F  C_  B
)  ->  ( (mrCls `  (SubMnd `  G )
) `  ran  F )  e.  (SubMnd `  G
) )
5635, 38, 55syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )  e.  (SubMnd `  G )
)
572oppgsubm 15150 . . . . . . . . 9  |-  (SubMnd `  G )  =  (SubMnd `  O )
5856, 57syl6eleq 2525 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )  e.  (SubMnd `  O )
)
59 eqid 2435 . . . . . . . . . 10  |-  ( +g  `  O )  =  ( +g  `  O )
6059submcl 14745 . . . . . . . . 9  |-  ( ( ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  e.  (SubMnd `  O )  /\  x  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  /\  y  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )  -> 
( x ( +g  `  O ) y )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
61603expb 1154 . . . . . . . 8  |-  ( ( ( (mrCls `  (SubMnd `  G ) ) `  ran  F )  e.  (SubMnd `  O )  /\  (
x  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F )  /\  y  e.  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) ) )  -> 
( x ( +g  `  O ) y )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
6258, 61sylan 458 . . . . . . 7  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  ( x  e.  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )  /\  y  e.  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) ) )  -> 
( x ( +g  `  O ) y )  e.  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
63 gsumzoppg.c . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
6463adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  F  C_  ( Z `  ran  F ) )
65 gsumzoppg.z . . . . . . . . . . . . . 14  |-  Z  =  (Cntz `  G )
66 eqid 2435 . . . . . . . . . . . . . 14  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
6765, 36, 66cntzspan 15452 . . . . . . . . . . . . 13  |-  ( ( G  e.  Mnd  /\  ran  F  C_  ( Z `  ran  F ) )  ->  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  e. CMnd )
6831, 64, 67syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  e. CMnd )
6966, 65submcmn2 15450 . . . . . . . . . . . . 13  |-  ( ( (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 ) ) ) )
7056, 69syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )  e. CMnd  <->  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) 
C_  ( Z `  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) ) ) )
7168, 70mpbid 202 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )  C_  ( Z `  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) ) )
7271sselda 3340 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  x  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )  ->  x  e.  ( Z `  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) ) )
73 eqid 2435 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
7473, 65cntzi 15120 . . . . . . . . . 10  |-  ( ( x  e.  ( Z `
 ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)  /\  y  e.  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )  ->  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) )
7572, 74sylan 458 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  x  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )  /\  y  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  ->  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
7673, 2, 59oppgplus 15137 . . . . . . . . 9  |-  ( x ( +g  `  O
) y )  =  ( y ( +g  `  G ) x )
7775, 76syl6reqr 2486 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  x  e.  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )  /\  y  e.  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  ->  ( x
( +g  `  O ) y )  =  ( x ( +g  `  G
) y ) )
7877anasss 629 . . . . . . 7  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  ( x  e.  ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )  /\  y  e.  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) ) )  -> 
( x ( +g  `  O ) y )  =  ( x ( +g  `  G ) y ) )
7924, 54, 62, 78seqfeq4 11364 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  (  seq  1
( ( +g  `  O
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  =  (  seq  1
( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
802, 32oppgbas 15139 . . . . . . 7  |-  B  =  ( Base `  O
)
81 eqid 2435 . . . . . . 7  |-  (Cntz `  O )  =  (Cntz `  O )
824adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  O  e.  Mnd )
835adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  A  e.  V
)
842, 65oppgcntz 15152 . . . . . . . 8  |-  ( Z `
 ran  F )  =  ( (Cntz `  O ) `  ran  F )
8564, 84syl6sseq 3386 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  F  C_  (
(Cntz `  O ) `  ran  F ) )
86 f1ofo 5673 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )
87 forn 5648 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
8886, 87syl 16 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  ran  f  =  ( `' F " ( _V  \  {  .0.  } ) ) )
8988ad2antll 710 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
9015, 89syl5sseqr 3389 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  ran  f )
91 eqid 2435 . . . . . . 7  |-  ( `' ( F  o.  f
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  o.  f ) " ( _V  \  {  .0.  }
) )
9280, 7, 59, 81, 82, 83, 25, 85, 22, 49, 90, 91gsumval3 15506 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( O  gsumg  F )  =  (  seq  1
( ( +g  `  O
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
9332, 6, 73, 65, 31, 83, 25, 64, 22, 49, 90, 91gsumval3 15506 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( G  gsumg  F )  =  (  seq  1
( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
9479, 92, 933eqtr4d 2477 . . . . 5  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( O  gsumg  F )  =  ( G  gsumg  F ) )
9594expr 599 . . . 4  |-  ( (
ph  /\  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )  ->  (
f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  ( O  gsumg  F )  =  ( G  gsumg  F ) ) )
9695exlimdv 1646 . . 3  |-  ( (
ph  /\  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )  ->  ( E. f  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) )  -> 
( O  gsumg  F )  =  ( G  gsumg  F ) ) )
9796expimpd 587 . 2  |-  ( ph  ->  ( ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )  ->  ( O  gsumg  F )  =  ( G  gsumg  F ) ) )
98 gsumzoppg.n . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
99 fz1f1o 12496 . . 3  |-  ( ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  \/  ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) ) )
10098, 99syl 16 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  \/  ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) ) )
10121, 97, 100mpjaod 371 1  |-  ( ph  ->  ( O  gsumg  F )  =  ( G  gsumg  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    C_ wss 3312   (/)c0 3620   {csn 3806    e. cmpt 4258   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873    o. ccom 4874    Fn wfn 5441   -->wf 5442   -1-1->wf1 5443   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   Fincfn 7101   1c1 8983   NNcn 9992   ZZ>=cuz 10480   ...cfz 11035    seq cseq 11315   #chash 11610   Basecbs 13461   ↾s cress 13462   +g cplusg 13521   0gc0g 13715    gsumg cgsu 13716  Moorecmre 13799  mrClscmrc 13800  ACScacs 13802   Mndcmnd 14676  SubMndcsubmnd 14729  Cntzccntz 15106  oppgcoppg 15133  CMndccmn 15404
This theorem is referenced by:  gsumzinv  15532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-cntz 15108  df-oppg 15134  df-cmn 15406
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