Theorem mgmlrid 13699

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 2238	. . . 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 13697	. . . 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 6093	. . . . 5 |- ( x = X -> ( .0. .+ x ) = ( .0. .+ X ) )
10		id 19	. . . . 5 |- ( x = X -> x = X )
11	9, 10	eqeq12d 2253	. . . 4 |- ( x = X -> ( ( .0. .+ x ) = x <-> ( .0. .+ X ) = X ) )
12		oveq1 6092	. . . . 5 |- ( x = X -> ( x .+ .0. ) = ( X .+ .0. ) )
13	12, 10	eqeq12d 2253	. . . 4 |- ( x = X -> ( ( x .+ .0. ) = x <-> ( X .+ .0. ) = X ) )
14	11, 13	anbi12d 477	. . 3 |- ( x = X -> ( ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) <-> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) ) )
15	14	rspccva 2928	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +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-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-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-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-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  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:  mndlrid  13747