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Theorem gsumval1 14708
Description: Value of the group sum operation when every element being summed is an identity of  G. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
gsumval1.b  |-  B  =  ( Base `  G
)
gsumval1.z  |-  .0.  =  ( 0g `  G )
gsumval1.p  |-  .+  =  ( +g  `  G )
gsumval1.o  |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) }
gsumval1.g  |-  ( ph  ->  G  e.  V )
gsumval1.a  |-  ( ph  ->  A  e.  W )
gsumval1.f  |-  ( ph  ->  F : A --> O )
Assertion
Ref Expression
gsumval1  |-  ( ph  ->  ( G  gsumg  F )  =  .0.  )
Distinct variable groups:    x, y, B    x,  .+ , y
Allowed substitution hints:    ph( x, y)    A( x, y)    F( x, y)    G( x, y)    O( x, y)    V( x, y)    W( x, y)    .0. ( x, y)

Proof of Theorem gsumval1
Dummy variables  f  m  n  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval1.b . . 3  |-  B  =  ( Base `  G
)
2 gsumval1.z . . 3  |-  .0.  =  ( 0g `  G )
3 gsumval1.p . . 3  |-  .+  =  ( +g  `  G )
4 gsumval1.o . . 3  |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) }
5 eqidd 2390 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  O
) )  =  ( `' F " ( _V 
\  O ) ) )
6 gsumval1.g . . 3  |-  ( ph  ->  G  e.  V )
7 gsumval1.a . . 3  |-  ( ph  ->  A  e.  W )
8 gsumval1.f . . . 4  |-  ( ph  ->  F : A --> O )
9 ssrab2 3373 . . . . 5  |-  { x  e.  B  |  A. y  e.  B  (
( x  .+  y
)  =  y  /\  ( y  .+  x
)  =  y ) }  C_  B
104, 9eqsstri 3323 . . . 4  |-  O  C_  B
11 fss 5541 . . . 4  |-  ( ( F : A --> O  /\  O  C_  B )  ->  F : A --> B )
128, 10, 11sylancl 644 . . 3  |-  ( ph  ->  F : A --> B )
131, 2, 3, 4, 5, 6, 7, 12gsumval 14704 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ...
,  ( iota z E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  z  =  (  seq  m (  .+  ,  F ) `  n
) ) ) ,  ( iota z E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  O
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  O ) )  /\  z  =  (  seq  1 (  .+  ,  ( F  o.  f ) ) `  ( # `  ( `' F " ( _V 
\  O ) ) ) ) ) ) ) ) )
14 frn 5539 . . 3  |-  ( F : A --> O  ->  ran  F  C_  O )
15 iftrue 3690 . . 3  |-  ( ran 
F  C_  O  ->  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ...
,  ( iota z E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  z  =  (  seq  m (  .+  ,  F ) `  n
) ) ) ,  ( iota z E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  O
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  O ) )  /\  z  =  (  seq  1 (  .+  ,  ( F  o.  f ) ) `  ( # `  ( `' F " ( _V 
\  O ) ) ) ) ) ) ) )  =  .0.  )
168, 14, 153syl 19 . 2  |-  ( ph  ->  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ...
,  ( iota z E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  z  =  (  seq  m (  .+  ,  F ) `  n
) ) ) ,  ( iota z E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  O
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  O ) )  /\  z  =  (  seq  1 (  .+  ,  ( F  o.  f ) ) `  ( # `  ( `' F " ( _V 
\  O ) ) ) ) ) ) ) )  =  .0.  )
1713, 16eqtrd 2421 1  |-  ( ph  ->  ( G  gsumg  F )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   A.wral 2651   E.wrex 2652   {crab 2655   _Vcvv 2901    \ cdif 3262    C_ wss 3265   ifcif 3684   `'ccnv 4819   ran crn 4821   "cima 4823    o. ccom 4824   iotacio 5358   -->wf 5392   -1-1-onto->wf1o 5395   ` cfv 5396  (class class class)co 6022   1c1 8926   ZZ>=cuz 10422   ...cfz 10977    seq cseq 11252   #chash 11547   Basecbs 13398   +g cplusg 13458   0gc0g 13652    gsumg cgsu 13653
This theorem is referenced by:  gsum0  14709  gsumz  14710  gsumval2  14712
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 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-recs 6571  df-rdg 6606  df-seq 11253  df-gsum 13657
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