Theorem lidrideqd 18353

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	⊢ (𝜑 → 𝐿 ∈ 𝐵)
lidrideqd.r	⊢ (𝜑 → 𝑅 ∈ 𝐵)
lidrideqd.li	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)
lidrideqd.ri	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)

Assertion

Ref	Expression
lidrideqd	⊢ (𝜑 → 𝐿 = 𝑅)

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem lidrideqd

Step	Hyp	Ref	Expression
1		oveq1 7282	. . . 4 ⊢ (𝑥 = 𝐿 → (𝑥 + 𝑅) = (𝐿 + 𝑅))
2		id 22	. . . 4 ⊢ (𝑥 = 𝐿 → 𝑥 = 𝐿)
3	1, 2	eqeq12d 2754	. . 3 ⊢ (𝑥 = 𝐿 → ((𝑥 + 𝑅) = 𝑥 ↔ (𝐿 + 𝑅) = 𝐿))
4		lidrideqd.ri	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)
5		lidrideqd.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ 𝐵)
6	3, 4, 5	rspcdva 3562	. 2 ⊢ (𝜑 → (𝐿 + 𝑅) = 𝐿)
7		oveq2 7283	. . . 4 ⊢ (𝑥 = 𝑅 → (𝐿 + 𝑥) = (𝐿 + 𝑅))
8		id 22	. . . 4 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
9	7, 8	eqeq12d 2754	. . 3 ⊢ (𝑥 = 𝑅 → ((𝐿 + 𝑥) = 𝑥 ↔ (𝐿 + 𝑅) = 𝑅))
10		lidrideqd.li	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)
11		lidrideqd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
12	9, 10, 11	rspcdva 3562	. 2 ⊢ (𝜑 → (𝐿 + 𝑅) = 𝑅)
13	6, 12	eqtr3d 2780	1 ⊢ (𝜑 → 𝐿 = 𝑅)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∀wral 3064  (class class class)co 7275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  lidrididd  18354