Theorem 0g0 13522

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 4221	. . 3 |- (/) e. _V
2		base0 13195	. . . 4 |- (/) = ( Base ` (/) )
3		eqid 2231	. . . 4 |- ( +g ` (/) ) = ( +g ` (/) )
4		eqid 2231	. . . 4 |- ( 0g ` (/) ) = ( 0g ` (/) )
5	2, 3, 4	grpidvalg 13519	. . 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 3500	. . . . . 6 |- -. e e. (/)
8	7	intnanr 938	. . . . 5 |- -. ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) )
9	8	nex 1549	. . . 4 |- -. E. e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) )
10		euex 2109	. . . 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 668	. . 3 |- -. E! e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) )
12		iotanul 5309	. . 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 2253	1 |- (/) = ( 0g ` (/) )

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1398   E.wex 1541   E!weu 2079   e. wcel 2202   A.wral 2511   _Vcvv 2803   (/)c0 3496   iotacio 5291   ` cfv 5333  (class class class)co 6028   +g cplusg 13223   0gc0g 13402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151  df-0g 13404

This theorem is referenced by: (None)