Theorem 0g0 12661

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 4125	. . 3 |- (/) e. _V
2		base0 12478	. . . 4 |- (/) = ( Base ` (/) )
3		eqid 2175	. . . 4 |- ( +g ` (/) ) = ( +g ` (/) )
4		eqid 2175	. . . 4 |- ( 0g ` (/) ) = ( 0g ` (/) )
5	2, 3, 4	grpidvalg 12658	. . 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 3424	. . . . . 6 |- -. e e. (/)
8	7	intnanr 930	. . . . 5 |- -. ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) )
9	8	nex 1498	. . . 4 |- -. E. e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) )
10		euex 2054	. . . 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 662	. . 3 |- -. E! e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) )
12		iotanul 5185	. . 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 2197	1 |- (/) = ( 0g ` (/) )

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1353   E.wex 1490   E!weu 2024   e. wcel 2146   A.wral 2453   _Vcvv 2735   (/)c0 3420   iotacio 5168   ` cfv 5208  (class class class)co 5865   +g cplusg 12493   0gc0g 12627

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-riota 5821  df-ov 5868  df-inn 8893  df-ndx 12432  df-slot 12433  df-base 12435  df-0g 12629

This theorem is referenced by: (None)