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Theorem 0g0 15747
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 14525 . . 3  |-  (/)  =  (
Base `  (/) )
2 eqid 2467 . . 3  |-  ( +g  `  (/) )  =  ( +g  `  (/) )
3 eqid 2467 . . 3  |-  ( 0g
`  (/) )  =  ( 0g `  (/) )
41, 2, 3grpidval 15745 . 2  |-  ( 0g
`  (/) )  =  ( iota e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) ) )
5 noel 3789 . . . . . 6  |-  -.  e  e.  (/)
65intnanr 913 . . . . 5  |-  -.  (
e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
76nex 1610 . . . 4  |-  -.  E. e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
8 euex 2303 . . . 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 5564 . . 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 2497 1  |-  (/)  =  ( 0g `  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275   A.wral 2814   (/)c0 3785   iotacio 5547   ` cfv 5586  (class class class)co 6282   +g cplusg 14551   0gc0g 14691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-slot 14490  df-base 14491  df-0g 14693
This theorem is referenced by:  frmd0  15851  rngidval  16945
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