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Mirrors > Home > MPE Home > Th. List > lidrideqd | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
lidrideqd | ⊢ (𝜑 → 𝐿 = 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7156 | . . . 4 ⊢ (𝑥 = 𝐿 → (𝑥 + 𝑅) = (𝐿 + 𝑅)) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝐿 → 𝑥 = 𝐿) | |
3 | 1, 2 | eqeq12d 2836 | . . 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 3622 | . 2 ⊢ (𝜑 → (𝐿 + 𝑅) = 𝐿) |
7 | oveq2 7157 | . . . 4 ⊢ (𝑥 = 𝑅 → (𝐿 + 𝑥) = (𝐿 + 𝑅)) | |
8 | id 22 | . . . 4 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅) | |
9 | 7, 8 | eqeq12d 2836 | . . 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 3622 | . 2 ⊢ (𝜑 → (𝐿 + 𝑅) = 𝑅) |
13 | 6, 12 | eqtr3d 2857 | 1 ⊢ (𝜑 → 𝐿 = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∀wral 3137 (class class class)co 7149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-iota 6307 df-fv 6356 df-ov 7152 |
This theorem is referenced by: lidrididd 17875 |
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