Theorem lidrideqd 12635

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 5860	. . . 4 |- ( x = L -> ( x .+ R ) = ( L .+ R ) )
2		id 19	. . . 4 |- ( x = L -> x = L )
3	1, 2	eqeq12d 2185	. . 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 2839	. 2 |- ( ph -> ( L .+ R ) = L )
7		oveq2 5861	. . . 4 |- ( x = R -> ( L .+ x ) = ( L .+ R ) )
8		id 19	. . . 4 |- ( x = R -> x = R )
9	7, 8	eqeq12d 2185	. . 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 2839	. 2 |- ( ph -> ( L .+ R ) = R )
13	6, 12	eqtr3d 2205	1 |- ( ph -> L = R )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   A.wral 2448  (class class class)co 5853

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-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  lidrididd  12636