Theorem lidrideqd 13288

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 5964	. . . 4 ⊢ (𝑥 = 𝐿 → (𝑥 + 𝑅) = (𝐿 + 𝑅))
2		id 19	. . . 4 ⊢ (𝑥 = 𝐿 → 𝑥 = 𝐿)
3	1, 2	eqeq12d 2221	. . 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 2886	. 2 ⊢ (𝜑 → (𝐿 + 𝑅) = 𝐿)
7		oveq2 5965	. . . 4 ⊢ (𝑥 = 𝑅 → (𝐿 + 𝑥) = (𝐿 + 𝑅))
8		id 19	. . . 4 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅)
9	7, 8	eqeq12d 2221	. . 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 2886	. 2 ⊢ (𝜑 → (𝐿 + 𝑅) = 𝑅)
13	6, 12	eqtr3d 2241	1 ⊢ (𝜑 → 𝐿 = 𝑅)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2177  ∀wral 2485  (class class class)co 5957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-iota 5241  df-fv 5288  df-ov 5960

This theorem is referenced by:  lidrididd  13289