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Theorem 0g0 16212
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 14880 . . 3  |-  (/)  =  (
Base `  (/) )
2 eqid 2402 . . 3  |-  ( +g  `  (/) )  =  ( +g  `  (/) )
3 eqid 2402 . . 3  |-  ( 0g
`  (/) )  =  ( 0g `  (/) )
41, 2, 3grpidval 16209 . 2  |-  ( 0g
`  (/) )  =  ( iota e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) ) )
5 noel 3741 . . . . . 6  |-  -.  e  e.  (/)
65intnanr 916 . . . . 5  |-  -.  (
e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
76nex 1648 . . . 4  |-  -.  E. e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
8 euex 2264 . . . 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 176 . . 3  |-  -.  E! e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
10 iotanul 5547 . . 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 2432 1  |-  (/)  =  ( 0g `  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   E!weu 2238   A.wral 2753   (/)c0 3737   iotacio 5530   ` cfv 5568  (class class class)co 6277   +g cplusg 14907   0gc0g 15052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-slot 14843  df-base 14844  df-0g 15054
This theorem is referenced by:  frmd0  16350  ringidval  17473
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