Theorem lidrideqd 12612

Description: If there is a left and right identity element for any binary operation (group operation) .+, both identity elements are equal. Generalization of statement in [Lang] p. 3: it is sufficient that "e" is a left identity element and "e`" is a right identity element instead of both being (two-sided) identity elements. (Contributed by AV, 26-Dec-2023.)

Hypotheses

Ref	Expression
lidrideqd.l	|- ( ph -> L e. B )
lidrideqd.r	|- ( ph -> R e. B )
lidrideqd.li	|- ( ph -> A. x e. B ( L .+ x ) = x )
lidrideqd.ri	|- ( ph -> A. x e. B ( x .+ R ) = x )

Assertion

Ref	Expression
lidrideqd	|- ( ph -> L = R )

Distinct variable groups:   x, B   x, L   x, R   x, .+

Allowed substitution hint:   ph( x)

Proof of Theorem lidrideqd

Step	Hyp	Ref	Expression
1		oveq1 5849	. . . 4 |- ( x = L -> ( x .+ R ) = ( L .+ R ) )
2		id 19	. . . 4 |- ( x = L -> x = L )
3	1, 2	eqeq12d 2180	. . 3 |- ( x = L -> ( ( x .+ R ) = x <-> ( L .+ R ) = L ) )
4		lidrideqd.ri	. . 3 |- ( ph -> A. x e. B ( x .+ R ) = x )
5		lidrideqd.l	. . 3 |- ( ph -> L e. B )
6	3, 4, 5	rspcdva 2835	. 2 |- ( ph -> ( L .+ R ) = L )
7		oveq2 5850	. . . 4 |- ( x = R -> ( L .+ x ) = ( L .+ R ) )
8		id 19	. . . 4 |- ( x = R -> x = R )
9	7, 8	eqeq12d 2180	. . 3 |- ( x = R -> ( ( L .+ x ) = x <-> ( L .+ R ) = R ) )
10		lidrideqd.li	. . 3 |- ( ph -> A. x e. B ( L .+ x ) = x )
11		lidrideqd.r	. . 3 |- ( ph -> R e. B )
12	9, 10, 11	rspcdva 2835	. 2 |- ( ph -> ( L .+ R ) = R )
13	6, 12	eqtr3d 2200	1 |- ( ph -> L = R )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   A.wral 2444  (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-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  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:  lidrididd  12613