Theorem mgmlrid 12610

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 2165	. . . 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 12608	. . . 4 |- ( ph -> ( ( .0. e. B /\ A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) <-> .0. = .0. ) )
7	1, 6	mpbiri 167	. . 3 |- ( ph -> ( .0. e. B /\ A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) )
8	7	simprd 113	. 2 |- ( ph -> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) )
9		oveq2 5850	. . . . 5 |- ( x = X -> ( .0. .+ x ) = ( .0. .+ X ) )
10		id 19	. . . . 5 |- ( x = X -> x = X )
11	9, 10	eqeq12d 2180	. . . 4 |- ( x = X -> ( ( .0. .+ x ) = x <-> ( .0. .+ X ) = X ) )
12		oveq1 5849	. . . . 5 |- ( x = X -> ( x .+ .0. ) = ( X .+ .0. ) )
13	12, 10	eqeq12d 2180	. . . 4 |- ( x = X -> ( ( x .+ .0. ) = x <-> ( X .+ .0. ) = X ) )
14	11, 13	anbi12d 465	. . 3 |- ( x = X -> ( ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) <-> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) ) )
15	14	rspccva 2829	. 2 |- ( ( A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) )
16	8, 15	sylan 281	1 |- ( ( ph /\ X e. B ) -> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   ` cfv 5188  (class class class)co 5842   Basecbs 12394   +g cplusg 12457   0gc0g 12573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196  df-riota 5798  df-ov 5845  df-inn 8858  df-ndx 12397  df-slot 12398  df-base 12400  df-0g 12575

This theorem is referenced by: (None)