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Theorem gsum0 14556
Description: Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypothesis
Ref Expression
gsum0.z  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
gsum0  |-  ( G 
gsumg  (/) )  =  .0.

Proof of Theorem gsum0
Dummy variables  f 
g  m  n  o  w  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2358 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 gsum0.z . . 3  |-  .0.  =  ( 0g `  G )
3 eqid 2358 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2358 . . 3  |-  { x  e.  ( Base `  G
)  |  A. y  e.  ( Base `  G
) ( ( x ( +g  `  G
) y )  =  y  /\  ( y ( +g  `  G
) x )  =  y ) }  =  { x  e.  ( Base `  G )  | 
A. y  e.  (
Base `  G )
( ( x ( +g  `  G ) y )  =  y  /\  ( y ( +g  `  G ) x )  =  y ) }
5 id 19 . . 3  |-  ( G  e.  _V  ->  G  e.  _V )
6 0ex 4231 . . . 4  |-  (/)  e.  _V
76a1i 10 . . 3  |-  ( G  e.  _V  ->  (/)  e.  _V )
8 f0 5508 . . . 4  |-  (/) : (/) --> { x  e.  ( Base `  G )  |  A. y  e.  ( Base `  G ) ( ( x ( +g  `  G
) y )  =  y  /\  ( y ( +g  `  G
) x )  =  y ) }
98a1i 10 . . 3  |-  ( G  e.  _V  ->  (/) : (/) --> { x  e.  ( Base `  G )  |  A. y  e.  ( Base `  G ) ( ( x ( +g  `  G
) y )  =  y  /\  ( y ( +g  `  G
) x )  =  y ) } )
101, 2, 3, 4, 5, 7, 9gsumval1 14555 . 2  |-  ( G  e.  _V  ->  ( G  gsumg  (/) )  =  .0.  )
11 df-gsum 13504 . . . . 5  |-  gsumg  =  ( w  e. 
_V ,  f  e. 
_V  |->  [_ { x  e.  ( Base `  w
)  |  A. y  e.  ( Base `  w
) ( ( x ( +g  `  w
) y )  =  y  /\  ( y ( +g  `  w
) x )  =  y ) }  / 
o ]_ if ( ran  f  C_  o , 
( 0g `  w
) ,  if ( dom  f  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( dom  f  =  ( m ... n
)  /\  x  =  (  seq  m ( ( +g  `  w ) ,  f ) `  n ) ) ) ,  ( iota x E. g [. ( `' f " ( _V 
\  o ) )  /  y ]. (
g : ( 1 ... ( # `  y
) ) -1-1-onto-> y  /\  x  =  (  seq  1
( ( +g  `  w
) ,  ( f  o.  g ) ) `
 ( # `  y
) ) ) ) ) ) )
1211reldmmpt2 6042 . . . 4  |-  Rel  dom  gsumg
1312ovprc1 5973 . . 3  |-  ( -.  G  e.  _V  ->  ( G  gsumg  (/) )  =  (/) )
14 fvprc 5602 . . . 4  |-  ( -.  G  e.  _V  ->  ( 0g `  G )  =  (/) )
152, 14syl5eq 2402 . . 3  |-  ( -.  G  e.  _V  ->  .0.  =  (/) )
1613, 15eqtr4d 2393 . 2  |-  ( -.  G  e.  _V  ->  ( G  gsumg  (/) )  =  .0.  )
1710, 16pm2.61i 156 1  |-  ( G 
gsumg  (/) )  =  .0.
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   E.wex 1541    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620   {crab 2623   _Vcvv 2864   [.wsbc 3067   [_csb 3157    \ cdif 3225    C_ wss 3228   (/)c0 3531   ifcif 3641   `'ccnv 4770   dom cdm 4771   ran crn 4772   "cima 4774    o. ccom 4775   iotacio 5299   -->wf 5333   -1-1-onto->wf1o 5336   ` cfv 5337  (class class class)co 5945   1c1 8828   ZZ>=cuz 10322   ...cfz 10874    seq cseq 11138   #chash 11430   Basecbs 13245   +g cplusg 13305   0gc0g 13499    gsumg cgsu 13500
This theorem is referenced by:  gsumwsubmcl  14560  gsumccat  14563  gsumwmhm  14566  gsumwspan  14567  frmdgsum  14583  frmdup1  14585  gsumwrev  14938  gsumconst  15308  mplmonmul  16307  mplcoe1  16308  mplcoe2  16310  gsumfsum  16545  tmdgsum  17880  xrge0gsumle  18441  xrge0tsms  18442  jensen  20394  xrge0tsmsd  23415  esumnul  23709  esumsn  23722  psgnunilem2  26741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-recs 6475  df-rdg 6510  df-seq 11139  df-gsum 13504
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