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Theorem gsumval1 14456
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 2284 . . 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 3258 . . . . 5  |-  { x  e.  B  |  A. y  e.  B  (
( x  .+  y
)  =  y  /\  ( y  .+  x
)  =  y ) }  C_  B
104, 9eqsstri 3208 . . . 4  |-  O  C_  B
11 fss 5397 . . . 4  |-  ( ( F : A --> O  /\  O  C_  B )  ->  F : A --> B )
128, 10, 11sylancl 643 . . 3  |-  ( ph  ->  F : A --> B )
131, 2, 3, 4, 5, 6, 7, 12gsumval 14452 . 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 5395 . . 3  |-  ( F : A --> O  ->  ran  F  C_  O )
15 iftrue 3571 . . 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 18 . 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 2315 1  |-  ( ph  ->  ( G  gsumg  F )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ifcif 3565   `'ccnv 4688   ran crn 4690   "cima 4692    o. ccom 4693   iotacio 5217   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   1c1 8738   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046   #chash 11337   Basecbs 13148   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401
This theorem is referenced by:  gsum0  14457  gsumz  14458  gsumval2  14460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-seq 11047  df-gsum 13405
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