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Mirrors > Home > MPE Home > Th. List > lidrididd | Structured version Visualization version GIF version |
Description: If there is a left and right identity element for any binary operation (group operation) +, the left identity element (and therefore also the right identity element according to lidrideqd 18695) is equal to the two-sided identity element. (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) |
lidrideqd.b | ⊢ 𝐵 = (Base‘𝐺) |
lidrideqd.p | ⊢ + = (+g‘𝐺) |
lidrididd.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
lidrididd | ⊢ (𝜑 → 𝐿 = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lidrideqd.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | lidrididd.o | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | lidrideqd.p | . 2 ⊢ + = (+g‘𝐺) | |
4 | lidrideqd.l | . 2 ⊢ (𝜑 → 𝐿 ∈ 𝐵) | |
5 | lidrideqd.li | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥) | |
6 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐿 + 𝑥) = (𝐿 + 𝑦)) | |
7 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
8 | 6, 7 | eqeq12d 2751 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐿 + 𝑥) = 𝑥 ↔ (𝐿 + 𝑦) = 𝑦)) |
9 | 8 | rspcv 3618 | . . 3 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥 → (𝐿 + 𝑦) = 𝑦)) |
10 | 5, 9 | mpan9 506 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐿 + 𝑦) = 𝑦) |
11 | lidrideqd.ri | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥) | |
12 | lidrideqd.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐵) | |
13 | 4, 12, 5, 11 | lidrideqd 18695 | . . . 4 ⊢ (𝜑 → 𝐿 = 𝑅) |
14 | oveq1 7438 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑅) = (𝑦 + 𝑅)) | |
15 | 14, 7 | eqeq12d 2751 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 + 𝑅) = 𝑥 ↔ (𝑦 + 𝑅) = 𝑦)) |
16 | 15 | rspcv 3618 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥 → (𝑦 + 𝑅) = 𝑦)) |
17 | oveq2 7439 | . . . . . . . . 9 ⊢ (𝐿 = 𝑅 → (𝑦 + 𝐿) = (𝑦 + 𝑅)) | |
18 | 17 | adantl 481 | . . . . . . . 8 ⊢ (((𝑦 + 𝑅) = 𝑦 ∧ 𝐿 = 𝑅) → (𝑦 + 𝐿) = (𝑦 + 𝑅)) |
19 | simpl 482 | . . . . . . . 8 ⊢ (((𝑦 + 𝑅) = 𝑦 ∧ 𝐿 = 𝑅) → (𝑦 + 𝑅) = 𝑦) | |
20 | 18, 19 | eqtrd 2775 | . . . . . . 7 ⊢ (((𝑦 + 𝑅) = 𝑦 ∧ 𝐿 = 𝑅) → (𝑦 + 𝐿) = 𝑦) |
21 | 20 | ex 412 | . . . . . 6 ⊢ ((𝑦 + 𝑅) = 𝑦 → (𝐿 = 𝑅 → (𝑦 + 𝐿) = 𝑦)) |
22 | 16, 21 | syl6com 37 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥 → (𝑦 ∈ 𝐵 → (𝐿 = 𝑅 → (𝑦 + 𝐿) = 𝑦))) |
23 | 22 | com23 86 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥 → (𝐿 = 𝑅 → (𝑦 ∈ 𝐵 → (𝑦 + 𝐿) = 𝑦))) |
24 | 11, 13, 23 | sylc 65 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑦 + 𝐿) = 𝑦)) |
25 | 24 | imp 406 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 + 𝐿) = 𝑦) |
26 | 1, 2, 3, 4, 10, 25 | ismgmid2 18694 | 1 ⊢ (𝜑 → 𝐿 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 |
This theorem is referenced by: (None) |
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