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Theorem gsumval1 14769
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 2436 . . 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 3420 . . . . 5  |-  { x  e.  B  |  A. y  e.  B  (
( x  .+  y
)  =  y  /\  ( y  .+  x
)  =  y ) }  C_  B
104, 9eqsstri 3370 . . . 4  |-  O  C_  B
11 fss 5591 . . . 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 14765 . 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 5589 . . 3  |-  ( F : A --> O  ->  ran  F  C_  O )
15 iftrue 3737 . . 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 2467 1  |-  ( ph  ->  ( G  gsumg  F )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948    \ cdif 3309    C_ wss 3312   ifcif 3731   `'ccnv 4869   ran crn 4871   "cima 4873    o. ccom 4874   iotacio 5408   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   1c1 8981   ZZ>=cuz 10478   ...cfz 11033    seq cseq 11313   #chash 11608   Basecbs 13459   +g cplusg 13519   0gc0g 13713    gsumg cgsu 13714
This theorem is referenced by:  gsum0  14770  gsumz  14771  gsumval2  14773
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2702  df-rex 2703  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-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  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-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-seq 11314  df-gsum 13718
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