Theorem mgmlrid 13461

Description: The identity element of a magma, if it exists, is a left and right identity. (Contributed by Mario Carneiro, 27-Dec-2014.)

Hypotheses

Ref	Expression
ismgmid.b	|- B = ( Base ` G )
ismgmid.o	|- .0. = ( 0g ` G )
ismgmid.p	|- .+ = ( +g ` G )
mgmidcl.e	|- ( ph -> E. e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )

Assertion

Ref	Expression
mgmlrid	|- ( ( ph /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) )

Distinct variable groups:   x, e, .+   .0. , e, x   B, e, x   e, G, x   x, X

Allowed substitution hints:   ph( x, e)   X( e)

Proof of Theorem mgmlrid

Step	Hyp	Ref	Expression
1		eqid 2231	. . . 4 |- .0. = .0.
2		ismgmid.b	. . . . 5 |- B = ( Base ` G )
3		ismgmid.o	. . . . 5 |- .0. = ( 0g ` G )
4		ismgmid.p	. . . . 5 |- .+ = ( +g ` G )
5		mgmidcl.e	. . . . 5 |- ( ph -> E. e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )
6	2, 3, 4, 5	ismgmid 13459	. . . 4 |- ( ph -> ( ( .0. e. B /\ A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) <-> .0. = .0. ) )
7	1, 6	mpbiri 168	. . 3 |- ( ph -> ( .0. e. B /\ A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) )
8	7	simprd 114	. 2 |- ( ph -> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
9		oveq2 6025	. . . . 5 |- ( x = X -> ( .0. .+ x ) = ( .0. .+ X ) )
10		id 19	. . . . 5 |- ( x = X -> x = X )
11	9, 10	eqeq12d 2246	. . . 4 |- ( x = X -> ( ( .0. .+ x ) = x <-> ( .0. .+ X ) = X ) )
12		oveq1 6024	. . . . 5 |- ( x = X -> ( x .+ .0. ) = ( X .+ .0. ) )
13	12, 10	eqeq12d 2246	. . . 4 |- ( x = X -> ( ( x .+ .0. ) = x <-> ( X .+ .0. ) = X ) )
14	11, 13	anbi12d 473	. . 3 |- ( x = X -> ( ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) <-> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) ) )
15	14	rspccva 2909	. 2 |- ( ( A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) )
16	8, 15	sylan 283	1 |- ( ( ph /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338

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-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-0g 13340

This theorem is referenced by:  mndlrid  13516