Theorem mgmlrid 13211

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 2205	. . . 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 13209	. . . 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 5952	. . . . 5 |- ( x = X -> ( .0. .+ x ) = ( .0. .+ X ) )
10		id 19	. . . . 5 |- ( x = X -> x = X )
11	9, 10	eqeq12d 2220	. . . 4 |- ( x = X -> ( ( .0. .+ x ) = x <-> ( .0. .+ X ) = X ) )
12		oveq1 5951	. . . . 5 |- ( x = X -> ( x .+ .0. ) = ( X .+ .0. ) )
13	12, 10	eqeq12d 2220	. . . 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 2876	. 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 1373   e. wcel 2176   A.wral 2484   E.wrex 2485   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   0gc0g 13088

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-inn 9037  df-ndx 12835  df-slot 12836  df-base 12838  df-0g 13090

This theorem is referenced by:  mndlrid  13266