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Theorem gsumval1 14472
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 2297 . . 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 3271 . . . . 5  |-  { x  e.  B  |  A. y  e.  B  (
( x  .+  y
)  =  y  /\  ( y  .+  x
)  =  y ) }  C_  B
104, 9eqsstri 3221 . . . 4  |-  O  C_  B
11 fss 5413 . . . 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 14468 . 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 5411 . . 3  |-  ( F : A --> O  ->  ran  F  C_  O )
15 iftrue 3584 . . 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 2328 1  |-  ( ph  ->  ( G  gsumg  F )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   ifcif 3578   `'ccnv 4704   ran crn 4706   "cima 4708    o. ccom 4709   iotacio 5233   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   1c1 8754   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062   #chash 11353   Basecbs 13164   +g cplusg 13224   0gc0g 13416    gsumg cgsu 13417
This theorem is referenced by:  gsum0  14473  gsumz  14474  gsumval2  14476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-seq 11063  df-gsum 13421
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