Theorem mgmidmo 17259

Description: A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by NM, 17-Jun-2017.)

Assertion

Ref	Expression
mgmidmo	|- E* u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x )

Distinct variable groups:   x, u, B   u, .+ , x

Proof of Theorem mgmidmo

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) -> ( u .+ x ) = x )
2	1	ralimi 2952	. . . 4 |- ( A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) -> A. x e. B ( u .+ x ) = x )
3		simpr 477	. . . . 5 |- ( ( ( w .+ x ) = x /\ ( x .+ w ) = x ) -> ( x .+ w ) = x )
4	3	ralimi 2952	. . . 4 |- ( A. x e. B ( ( w .+ x ) = x /\ ( x .+ w ) = x ) -> A. x e. B ( x .+ w ) = x )
5		oveq1 6657	. . . . . . . . 9 |- ( x = u -> ( x .+ w ) = ( u .+ w ) )
6		id 22	. . . . . . . . 9 |- ( x = u -> x = u )
7	5, 6	eqeq12d 2637	. . . . . . . 8 |- ( x = u -> ( ( x .+ w ) = x <-> ( u .+ w ) = u ) )
8	7	rspcva 3307	. . . . . . 7 |- ( ( u e. B /\ A. x e. B ( x .+ w ) = x ) -> ( u .+ w ) = u )
9		oveq2 6658	. . . . . . . . 9 |- ( x = w -> ( u .+ x ) = ( u .+ w ) )
10		id 22	. . . . . . . . 9 |- ( x = w -> x = w )
11	9, 10	eqeq12d 2637	. . . . . . . 8 |- ( x = w -> ( ( u .+ x ) = x <-> ( u .+ w ) = w ) )
12	11	rspcva 3307	. . . . . . 7 |- ( ( w e. B /\ A. x e. B ( u .+ x ) = x ) -> ( u .+ w ) = w )
13	8, 12	sylan9req 2677	. . . . . 6 |- ( ( ( u e. B /\ A. x e. B ( x .+ w ) = x ) /\ ( w e. B /\ A. x e. B ( u .+ x ) = x ) ) -> u = w )
14	13	an42s 870	. . . . 5 |- ( ( ( u e. B /\ w e. B ) /\ ( A. x e. B ( u .+ x ) = x /\ A. x e. B ( x .+ w ) = x ) ) -> u = w )
15	14	ex 450	. . . 4 |- ( ( u e. B /\ w e. B ) -> ( ( A. x e. B ( u .+ x ) = x /\ A. x e. B ( x .+ w ) = x ) -> u = w ) )
16	2, 4, 15	syl2ani 688	. . 3 |- ( ( u e. B /\ w e. B ) -> ( ( A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ A. x e. B ( ( w .+ x ) = x /\ ( x .+ w ) = x ) ) -> u = w ) )
17	16	rgen2a 2977	. 2 |- A. u e. B A. w e. B ( ( A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ A. x e. B ( ( w .+ x ) = x /\ ( x .+ w ) = x ) ) -> u = w )
18		oveq1 6657	. . . . . 6 |- ( u = w -> ( u .+ x ) = ( w .+ x ) )
19	18	eqeq1d 2624	. . . . 5 |- ( u = w -> ( ( u .+ x ) = x <-> ( w .+ x ) = x ) )
20		oveq2 6658	. . . . . 6 |- ( u = w -> ( x .+ u ) = ( x .+ w ) )
21	20	eqeq1d 2624	. . . . 5 |- ( u = w -> ( ( x .+ u ) = x <-> ( x .+ w ) = x ) )
22	19, 21	anbi12d 747	. . . 4 |- ( u = w -> ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) <-> ( ( w .+ x ) = x /\ ( x .+ w ) = x ) ) )
23	22	ralbidv 2986	. . 3 |- ( u = w -> ( A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) <-> A. x e. B ( ( w .+ x ) = x /\ ( x .+ w ) = x ) ) )
24	23	rmo4 3399	. 2 |- ( E* u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) <-> A. u e. B A. w e. B ( ( A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) /\ A. x e. B ( ( w .+ x ) = x /\ ( x .+ w ) = x ) ) -> u = w ) )
25	17, 24	mpbir 221	1 |- E* u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E*wrmo 2915  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  ismgmid  17264  mndideu  17304