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Theorem 0g0 14402
Description: The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)
Assertion
Ref Expression
0g0  |-  (/)  =  ( 0g `  (/) )

Proof of Theorem 0g0
Dummy variables  e  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 base0 13201 . . 3  |-  (/)  =  (
Base `  (/) )
2 eqid 2296 . . 3  |-  ( +g  `  (/) )  =  ( +g  `  (/) )
3 eqid 2296 . . 3  |-  ( 0g
`  (/) )  =  ( 0g `  (/) )
41, 2, 3grpidval 14400 . 2  |-  ( 0g
`  (/) )  =  ( iota e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) ) )
5 noel 3472 . . . . . 6  |-  -.  e  e.  (/)
65intnanr 881 . . . . 5  |-  -.  (
e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
76nex 1545 . . . 4  |-  -.  E. e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
8 euex 2179 . . . 4  |-  ( E! e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )  ->  E. e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) ) )
97, 8mto 167 . . 3  |-  -.  E! e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
10 iotanul 5250 . . 3  |-  ( -.  E! e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )  -> 
( iota e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) ) )  =  (/) )
119, 10ax-mp 8 . 2  |-  ( iota e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) ) )  =  (/)
124, 11eqtr2i 2317 1  |-  (/)  =  ( 0g `  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156   A.wral 2556   (/)c0 3468   iotacio 5233   ` cfv 5271  (class class class)co 5874   +g cplusg 13224   0gc0g 13416
This theorem is referenced by:  frmd0  14498  rngidval  15359
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
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-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-slot 13168  df-base 13169  df-0g 13420
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