Theorem 0g0 13696

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.

Step	Hyp	Ref	Expression
1		0ex 4260	. . 3 |- (/) e. _V
2		base0 13402	. . . 4 |- (/) = ( Base ` (/) )
3		eqid 2238	. . . 4 |- ( +g ` (/) ) = ( +g ` (/) )
4		eqid 2238	. . . 4 |- ( 0g ` (/) ) = ( 0g ` (/) )
5	2, 3, 4	grpidvalg 13693	. . 3 |- ( (/) e. _V -> ( 0g ` (/) ) = ( iota e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) ) ) )
6	1, 5	ax-mp 5	. 2 |- ( 0g ` (/) ) = ( iota e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) ) )
7		noel 3525	. . . . . 6 |- -. e e. (/)
8	7	intnanr 942	. . . . 5 |- -. ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) )
9	8	nex 1553	. . . 4 |- -. E. e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) )
10		euex 2116	. . . 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 ) ) )
11	9, 10	mto 672	. . 3 |- -. E! e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) )
12		iotanul 5353	. . 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 ) ) ) = (/) )
13	11, 12	ax-mp 5	. 2 |- ( iota e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) ) ) = (/)
14	6, 13	eqtr2i 2260	1 |- (/) = ( 0g ` (/) )

This proof depends on syntax axioms:   -. wn 3   /\ wa 104   = wceq 1402   E.wex 1545   E!weu 2086   e. wcel 2209   A.wral 2528   _Vcvv 2821   (/)c0 3520   iotacio 5335   ` cfv 5377  (class class class)co 6085   +g cplusg 13431   0gc0g 13610

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9305  df-ndx 13355  df-slot 13356  df-base 13358  df-0g 13612

This theorem is used by: (None)