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Theorem gsumzoppg 15466
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 15077 . . . . . . . 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 15079 . . . . . . . 8  |-  .0.  =  ( 0g `  O )
87gsumz 14708 . . . . . . 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 14708 . . . . . . 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 2422 . . . . 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 3310 . . . . . . 7  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
1615a1i 11 . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
1714, 16gsumcllem 15443 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
1817oveq2d 6036 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( O  gsumg  F )  =  ( O  gsumg  ( k  e.  A  |->  .0.  ) ) )
1917oveq2d 6036 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
2013, 18, 193eqtr4d 2429 . . 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 10453 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2422, 23syl6eleq 2477 . . . . . . 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 5531 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  F  Fn  A )
27 dffn4 5599 . . . . . . . . . . . 12  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
2826, 27sylib 189 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F : A -onto-> ran  F
)
29 fof 5593 . . . . . . . . . . 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 14692 . . . . . . . . . . . 12  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
34 acsmre 13804 . . . . . . . . . . . 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 2387 . . . . . . . . . . 11  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
37 frn 5537 . . . . . . . . . . . 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 13777 . . . . . . . . . 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 5539 . . . . . . . . . 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 5613 . . . . . . . . . . . 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 5164 . . . . . . . . . . . 12  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
45 fdm 5535 . . . . . . . . . . . . 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 3339 . . . . . . . . . . 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 5584 . . . . . . . . . . 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 5579 . . . . . . . . . 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 5540 . . . . . . . . 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 5809 . . . . . . 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 13763 . . . . . . . . . 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 15085 . . . . . . . . 9  |-  (SubMnd `  G )  =  (SubMnd `  O )
5856, 57syl6eleq 2477 . . . . . . . 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 2387 . . . . . . . . . 10  |-  ( +g  `  O )  =  ( +g  `  O )
6059submcl 14680 . . . . . . . . 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 2387 . . . . . . . . . . . . . 14  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
6765, 36, 66cntzspan 15387 . . . . . . . . . . . . 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 15385 . . . . . . . . . . . . 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 3291 . . . . . . . . . 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 2387 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
7473, 65cntzi 15055 . . . . . . . . . 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 15072 . . . . . . . . 9  |-  ( x ( +g  `  O
) y )  =  ( y ( +g  `  G ) x )
7775, 76syl6reqr 2438 . . . . . . . 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 11299 . . . . . 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 15074 . . . . . . 7  |-  B  =  ( Base `  O
)
81 eqid 2387 . . . . . . 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 15087 . . . . . . . 8  |-  ( Z `
 ran  F )  =  ( (Cntz `  O ) `  ran  F )
8564, 84syl6sseq 3337 . . . . . . 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 5621 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )
87 forn 5596 . . . . . . . . . 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 3340 . . . . . . 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 2387 . . . . . . 7  |-  ( `' ( F  o.  f
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  o.  f ) " ( _V  \  {  .0.  }
) )
9280, 7, 59, 81, 82, 83, 25, 85, 22, 49, 90, 91gsumval3 15441 . . . . . 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 15441 . . . . . 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 2429 . . . . 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 1643 . . 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 12431 . . 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 1547    = wceq 1649    e. wcel 1717   _Vcvv 2899    \ cdif 3260    C_ wss 3263   (/)c0 3571   {csn 3757    e. cmpt 4207   `'ccnv 4817   dom cdm 4818   ran crn 4819   "cima 4821    o. ccom 4822    Fn wfn 5389   -->wf 5390   -1-1->wf1 5391   -onto->wfo 5392   -1-1-onto->wf1o 5393   ` cfv 5394  (class class class)co 6020   Fincfn 7045   1c1 8924   NNcn 9932   ZZ>=cuz 10420   ...cfz 10975    seq cseq 11250   #chash 11545   Basecbs 13396   ↾s cress 13397   +g cplusg 13456   0gc0g 13650    gsumg cgsu 13651  Moorecmre 13734  mrClscmrc 13735  ACScacs 13737   Mndcmnd 14611  SubMndcsubmnd 14664  Cntzccntz 15041  oppgcoppg 15068  CMndccmn 15339
This theorem is referenced by:  gsumzinv  15467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-fzo 11066  df-seq 11251  df-hash 11546  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-0g 13654  df-gsum 13655  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-cntz 15043  df-oppg 15069  df-cmn 15341
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