Theorem lidrideqd 18622

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 7421	. . . 4 ⊢ (𝑥 = 𝐿 → (𝑥 + 𝑅) = (𝐿 + 𝑅))
2		id 22	. . . 4 ⊢ (𝑥 = 𝐿 → 𝑥 = 𝐿)
3	1, 2	eqeq12d 2744	. . 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 3609	. 2 ⊢ (𝜑 → (𝐿 + 𝑅) = 𝐿)
7		oveq2 7422	. . . 4 ⊢ (𝑥 = 𝑅 → (𝐿 + 𝑥) = (𝐿 + 𝑅))
8		id 22	. . . 4 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
9	7, 8	eqeq12d 2744	. . 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 3609	. 2 ⊢ (𝜑 → (𝐿 + 𝑅) = 𝑅)
13	6, 12	eqtr3d 2770	1 ⊢ (𝜑 → 𝐿 = 𝑅)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ∀wral 3057  (class class class)co 7414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417

This theorem is referenced by:  lidrididd  18623