MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumzoppgOLD Structured version   Unicode version

Theorem gsumzoppgOLD 16463
Description: The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of gsumzoppg 16462 as of 6-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
gsumzoppgOLD.b  |-  B  =  ( Base `  G
)
gsumzoppgOLD.0  |-  .0.  =  ( 0g `  G )
gsumzoppgOLD.z  |-  Z  =  (Cntz `  G )
gsumzoppgOLD.o  |-  O  =  (oppg
`  G )
gsumzoppgOLD.g  |-  ( ph  ->  G  e.  Mnd )
gsumzoppgOLD.a  |-  ( ph  ->  A  e.  V )
gsumzoppgOLD.f  |-  ( ph  ->  F : A --> B )
gsumzoppgOLD.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzoppgOLD.n  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
Assertion
Ref Expression
gsumzoppgOLD  |-  ( ph  ->  ( O  gsumg  F )  =  ( G  gsumg  F ) )

Proof of Theorem gsumzoppgOLD
Dummy variables  f 
k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzoppgOLD.g . . . . . . . 8  |-  ( ph  ->  G  e.  Mnd )
2 gsumzoppgOLD.o . . . . . . . . 9  |-  O  =  (oppg
`  G )
32oppgmnd 15890 . . . . . . . 8  |-  ( G  e.  Mnd  ->  O  e.  Mnd )
41, 3syl 16 . . . . . . 7  |-  ( ph  ->  O  e.  Mnd )
5 gsumzoppgOLD.a . . . . . . 7  |-  ( ph  ->  A  e.  V )
6 gsumzoppgOLD.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
72, 6oppgid 15892 . . . . . . . 8  |-  .0.  =  ( 0g `  O )
87gsumz 15532 . . . . . . 7  |-  ( ( O  e.  Mnd  /\  A  e.  V )  ->  ( O  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
94, 5, 8syl2anc 661 . . . . . 6  |-  ( ph  ->  ( O  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
106gsumz 15532 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
111, 5, 10syl2anc 661 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
129, 11eqtr4d 2478 . . . . 5  |-  ( ph  ->  ( O  gsumg  ( k  e.  A  |->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
1312adantr 465 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( O  gsumg  ( k  e.  A  |->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
14 gsumzoppgOLD.f . . . . . 6  |-  ( ph  ->  F : A --> B )
15 ssid 3396 . . . . . . 7  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
1615a1i 11 . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
1714, 16gsumcllemOLD 16408 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
1817oveq2d 6128 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( O  gsumg  F )  =  ( O  gsumg  ( k  e.  A  |->  .0.  ) ) )
1917oveq2d 6128 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
2013, 18, 193eqtr4d 2485 . . 3  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( O  gsumg  F )  =  ( G  gsumg  F ) )
2120ex 434 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  ->  ( O  gsumg  F )  =  ( G  gsumg  F ) ) )
22 simprl 755 . . . . . . . 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 10917 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
2422, 23syl6eleq 2533 . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  F : A --> B )
26 ffn 5580 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  F  Fn  A )
27 dffn4 5647 . . . . . . . . . . . 12  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
2826, 27sylib 196 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F : A -onto-> ran  F
)
29 fof 5641 . . . . . . . . . . 11  |-  ( F : A -onto-> ran  F  ->  F : A --> ran  F
)
3025, 28, 293syl 20 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  F : A --> ran  F )
311adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  G  e.  Mnd )
32 gsumzoppgOLD.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  G
)
3332submacs 15514 . . . . . . . . . . . 12  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  B ) )
34 acsmre 14611 . . . . . . . . . . . 12  |-  ( (SubMnd `  G )  e.  (ACS
`  B )  -> 
(SubMnd `  G )  e.  (Moore `  B )
)
3531, 33, 343syl 20 . . . . . . . . . . 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 2443 . . . . . . . . . . 11  |-  (mrCls `  (SubMnd `  G ) )  =  (mrCls `  (SubMnd `  G ) )
37 frn 5586 . . . . . . . . . . . 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 14584 . . . . . . . . . 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 5588 . . . . . . . . . 10  |-  ( ( F : A --> ran  F  /\  ran  F  C_  (
(mrCls `  (SubMnd `  G
) ) `  ran  F ) )  ->  F : A --> ( (mrCls `  (SubMnd `  G ) ) `
 ran  F )
)
4130, 39, 40syl2anc 661 . . . . . . . . 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 5661 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) ) )
4342ad2antll 728 . . . . . . . . . . 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 5210 . . . . . . . . . . . 12  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
45 fdm 5584 . . . . . . . . . . . . 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 3425 . . . . . . . . . . 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 5632 . . . . . . . . . . 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 661 . . . . . . . . . 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 5627 . . . . . . . . . 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 5589 . . . . . . . . 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 661 . . . . . . . 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 5864 . . . . . . 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 14570 . . . . . . . . . 10  |-  ( ( (SubMnd `  G )  e.  (Moore `  B )  /\  ran  F  C_  B
)  ->  ( (mrCls `  (SubMnd `  G )
) `  ran  F )  e.  (SubMnd `  G
) )
5635, 38, 55syl2anc 661 . . . . . . . . 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 15898 . . . . . . . . 9  |-  (SubMnd `  G )  =  (SubMnd `  O )
5856, 57syl6eleq 2533 . . . . . . . 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 2443 . . . . . . . . . 10  |-  ( +g  `  O )  =  ( +g  `  O )
6059submcl 15502 . . . . . . . . 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 1188 . . . . . . . 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 471 . . . . . . 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 gsumzoppgOLD.c . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
6463adantr 465 . . . . . . . . . . . . 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 gsumzoppgOLD.z . . . . . . . . . . . . . 14  |-  Z  =  (Cntz `  G )
66 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Gs  ( (mrCls `  (SubMnd `  G
) ) `  ran  F ) )  =  ( Gs  ( (mrCls `  (SubMnd `  G ) ) `  ran  F ) )
6765, 36, 66cntzspan 16347 . . . . . . . . . . . . 13  |-  ( ( G  e.  Mnd  /\  ran  F  C_  ( Z `  ran  F ) )  ->  ( Gs  ( (mrCls `  (SubMnd `  G )
) `  ran  F ) )  e. CMnd )
6831, 64, 67syl2anc 661 . . . . . . . . . . . 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 16344 . . . . . . . . . . . . 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 210 . . . . . . . . . . 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 3377 . . . . . . . . . 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 2443 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
7473, 65cntzi 15868 . . . . . . . . . 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 471 . . . . . . . . 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 15885 . . . . . . . . 9  |-  ( x ( +g  `  O
) y )  =  ( y ( +g  `  G ) x )
7775, 76syl6reqr 2494 . . . . . . . 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 647 . . . . . . 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 11876 . . . . . 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 15887 . . . . . . 7  |-  B  =  ( Base `  O
)
81 eqid 2443 . . . . . . 7  |-  (Cntz `  O )  =  (Cntz `  O )
824adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  O  e.  Mnd )
835adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  A  e.  V
)
842, 65oppgcntz 15900 . . . . . . . 8  |-  ( Z `
 ran  F )  =  ( (Cntz `  O ) `  ran  F )
8564, 84syl6sseq 3423 . . . . . . 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 5669 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )
87 forn 5644 . . . . . . . . . 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 728 . . . . . . . 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 3426 . . . . . . 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 2443 . . . . . . 7  |-  ( `' ( F  o.  f
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  o.  f ) " ( _V  \  {  .0.  }
) )
9280, 7, 59, 81, 82, 83, 25, 85, 22, 49, 90, 91gsumval3OLD 16403 . . . . . 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, 91gsumval3OLD 16403 . . . . . 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 2485 . . . . 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 615 . . . 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 1690 . . 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 603 . 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 gsumzoppgOLD.n . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
99 fz1f1o 13208 . . 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 381 1  |-  ( ph  ->  ( O  gsumg  F )  =  ( G  gsumg  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2993    \ cdif 3346    C_ wss 3349   (/)c0 3658   {csn 3898    e. cmpt 4371   `'ccnv 4860   dom cdm 4861   ran crn 4862   "cima 4864    o. ccom 4865    Fn wfn 5434   -->wf 5435   -1-1->wf1 5436   -onto->wfo 5437   -1-1-onto->wf1o 5438   ` cfv 5439  (class class class)co 6112   Fincfn 7331   1c1 9304   NNcn 10343   ZZ>=cuz 10882   ...cfz 11458    seqcseq 11827   #chash 12124   Basecbs 14195   ↾s cress 14196   +g cplusg 14259   0gc0g 14399    gsumg cgsu 14400  Moorecmre 14541  mrClscmrc 14542  ACScacs 14544   Mndcmnd 15430  SubMndcsubmnd 15484  Cntzccntz 15854  oppgcoppg 15881  CMndccmn 16298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-fzo 11570  df-seq 11828  df-hash 12125  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-0g 14401  df-gsum 14402  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-submnd 15486  df-cntz 15856  df-oppg 15882  df-cmn 16300
This theorem is referenced by:  gsumzinvOLD  16465
  Copyright terms: Public domain W3C validator