Theorem mgmidmo 12603

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 108	. . . . 5 |- ( ( ( u .+ x ) = x /\ ( x .+ u ) = x ) -> ( u .+ x ) = x )
2	1	ralimi 2529	. . . 4 |- ( A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) -> A. x e. B ( u .+ x ) = x )
3		simpr 109	. . . . 5 |- ( ( ( w .+ x ) = x /\ ( x .+ w ) = x ) -> ( x .+ w ) = x )
4	3	ralimi 2529	. . . 4 |- ( A. x e. B ( ( w .+ x ) = x /\ ( x .+ w ) = x ) -> A. x e. B ( x .+ w ) = x )
5		oveq1 5849	. . . . . . . . 9 |- ( x = u -> ( x .+ w ) = ( u .+ w ) )
6		id 19	. . . . . . . . 9 |- ( x = u -> x = u )
7	5, 6	eqeq12d 2180	. . . . . . . 8 |- ( x = u -> ( ( x .+ w ) = x <-> ( u .+ w ) = u ) )
8	7	rspcva 2828	. . . . . . 7 |- ( ( u e. B /\ A. x e. B ( x .+ w ) = x ) -> ( u .+ w ) = u )
9		oveq2 5850	. . . . . . . . 9 |- ( x = w -> ( u .+ x ) = ( u .+ w ) )
10		id 19	. . . . . . . . 9 |- ( x = w -> x = w )
11	9, 10	eqeq12d 2180	. . . . . . . 8 |- ( x = w -> ( ( u .+ x ) = x <-> ( u .+ w ) = w ) )
12	11	rspcva 2828	. . . . . . 7 |- ( ( w e. B /\ A. x e. B ( u .+ x ) = x ) -> ( u .+ w ) = w )
13	8, 12	sylan9req 2220	. . . . . 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 579	. . . . 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 114	. . . 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 406	. . 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	rgen2 2552	. 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 5849	. . . . 5 |- ( u = w -> ( u .+ x ) = ( w .+ x ) )
19	18	eqeq1d 2174	. . . 4 |- ( u = w -> ( ( u .+ x ) = x <-> ( w .+ x ) = x ) )
20	19	ovanraleqv 5866	. . 3 |- ( u = w -> ( A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x ) <-> A. x e. B ( ( w .+ x ) = x /\ ( x .+ w ) = x ) ) )
21	20	rmo4 2919	. 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 ) )
22	17, 21	mpbir 145	1 |- E* u e. B A. x e. B ( ( u .+ x ) = x /\ ( x .+ u ) = x )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   E*wrmo 2447  (class class class)co 5842

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-ext 2147

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-rmo 2452  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  ismgmid  12608