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Theorem 0g0 14386
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 13185 . . 3  |-  (/)  =  (
Base `  (/) )
2 eqid 2283 . . 3  |-  ( +g  `  (/) )  =  ( +g  `  (/) )
3 eqid 2283 . . 3  |-  ( 0g
`  (/) )  =  ( 0g `  (/) )
41, 2, 3grpidval 14384 . 2  |-  ( 0g
`  (/) )  =  ( iota e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) ) )
5 noel 3459 . . . . . 6  |-  -.  e  e.  (/)
65intnanr 881 . . . . 5  |-  -.  (
e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
76nex 1542 . . . 4  |-  -.  E. e ( e  e.  (/)  /\  A. x  e.  (/)  ( ( e ( +g  `  (/) ) x )  =  x  /\  ( x ( +g  `  (/) ) e )  =  x ) )
8 euex 2166 . . . 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 5234 . . 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 2304 1  |-  (/)  =  ( 0g `  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   A.wral 2543   (/)c0 3455   iotacio 5217   ` cfv 5255  (class class class)co 5858   +g cplusg 13208   0gc0g 13400
This theorem is referenced by:  frmd0  14482  rngidval  15343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-slot 13152  df-base 13153  df-0g 13404
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