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	⊢ (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 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 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)
5		lidrideqd.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ 𝐵)
6	3, 4, 5	rspcdva 2839	. 2 ⊢ (𝜑 → (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 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)
11		lidrideqd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
12	9, 10, 11	rspcdva 2839	. 2 ⊢ (𝜑 → (𝐿 + 𝑅) = 𝑅)
13	6, 12	eqtr3d 2205	1 ⊢ (𝜑 → 𝐿 = 𝑅)

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  ∀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