Theorem 0g0 13404

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 4210	. . 3 |- (/) e. _V
2		base0 13077	. . . 4 |- (/) = ( Base ` (/) )
3		eqid 2229	. . . 4 |- ( +g ` (/) ) = ( +g ` (/) )
4		eqid 2229	. . . 4 |- ( 0g ` (/) ) = ( 0g ` (/) )
5	2, 3, 4	grpidvalg 13401	. . 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 3495	. . . . . 6 |- -. e e. (/)
8	7	intnanr 935	. . . . 5 |- -. ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) )
9	8	nex 1546	. . . 4 |- -. E. e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) )
10		euex 2107	. . . 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 666	. . 3 |- -. E! e ( e e. (/) /\ A. x e. (/) ( ( e ( +g ` (/) ) x ) = x /\ ( x ( +g ` (/) ) e ) = x ) )
12		iotanul 5293	. . 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 2251	1 |- (/) = ( 0g ` (/) )

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1395   E.wex 1538   E!weu 2077   e. wcel 2200   A.wral 2508   _Vcvv 2799   (/)c0 3491   iotacio 5275   ` cfv 5317  (class class class)co 6000   +g cplusg 13105   0gc0g 13284

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-riota 5953  df-ov 6003  df-inn 9107  df-ndx 13030  df-slot 13031  df-base 13033  df-0g 13286

This theorem is referenced by: (None)