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Theorem 0g0 16506
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 15162 . . 3  |-  (/)  =  (
Base `  (/) )
2 eqid 2451 . . 3  |-  ( +g  `  (/) )  =  ( +g  `  (/) )
3 eqid 2451 . . 3  |-  ( 0g
`  (/) )  =  ( 0g `  (/) )
41, 2, 3grpidval 16503 . 2  |-  ( 0g
`  (/) )  =  ( iota e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) ) )
5 noel 3735 . . . . . 6  |-  -.  e  e.  (/)
65intnanr 926 . . . . 5  |-  -.  (
e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
76nex 1678 . . . 4  |-  -.  E. e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
8 euex 2323 . . . 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 180 . . 3  |-  -.  E! e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
10 iotanul 5561 . . 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 5 . 2  |-  ( iota e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) ) )  =  (/)
124, 11eqtr2i 2474 1  |-  (/)  =  ( 0g `  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887   E!weu 2299   A.wral 2737   (/)c0 3731   iotacio 5544   ` cfv 5582  (class class class)co 6290   +g cplusg 15190   0gc0g 15338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-slot 15125  df-base 15126  df-0g 15340
This theorem is referenced by:  frmd0  16644  ringidval  17737
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