Theorem mgmlrid 12633

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 2170	. . . 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 12631	. . . 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 5861	. . . . 5 |- ( x = X -> ( .0. .+ x ) = ( .0. .+ X ) )
10		id 19	. . . . 5 |- ( x = X -> x = X )
11	9, 10	eqeq12d 2185	. . . 4 |- ( x = X -> ( ( .0. .+ x ) = x <-> ( .0. .+ X ) = X ) )
12		oveq1 5860	. . . . 5 |- ( x = X -> ( x .+ .0. ) = ( X .+ .0. ) )
13	12, 10	eqeq12d 2185	. . . 4 |- ( x = X -> ( ( x .+ .0. ) = x <-> ( X .+ .0. ) = X ) )
14	11, 13	anbi12d 470	. . 3 |- ( x = X -> ( ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) <-> ( ( .0. .+ X ) = X /\ ( X .+ .0. ) = X ) ) )
15	14	rspccva 2833	. 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 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   ` cfv 5198  (class class class)co 5853   Basecbs 12416   +g cplusg 12480   0gc0g 12596

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-riota 5809  df-ov 5856  df-inn 8879  df-ndx 12419  df-slot 12420  df-base 12422  df-0g 12598

This theorem is referenced by:  mndlrid  12670