Theorem lidrideqd 18862

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 7434	. . . 4 ⊢ (𝑥 = 𝐿 → (𝑥 + 𝑅) = (𝐿 + 𝑅))
2		id 23	. . . 4 ⊢ (𝑥 = 𝐿 → 𝑥 = 𝐿)
3	1, 2	eqeq12d 2779	. . 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 3580	. 2 ⊢ (𝜑 → (𝐿 + 𝑅) = 𝐿)
7		oveq2 7435	. . . 4 ⊢ (𝑥 = 𝑅 → (𝐿 + 𝑥) = (𝐿 + 𝑅))
8		id 23	. . . 4 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
9	7, 8	eqeq12d 2779	. . 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 3580	. 2 ⊢ (𝜑 → (𝐿 + 𝑅) = 𝑅)
13	6, 12	eqtr3d 2800	1 ⊢ (𝜑 → 𝐿 = 𝑅)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  ∀wral 3079  (class class class)co 7427

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-iota 6500  df-fv 6554  df-ov 7430

This theorem is used by:  lidrididd  18863