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| 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 18683) 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 7440 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐿 + 𝑥) = (𝐿 + 𝑦)) | |
| 7 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
| 8 | 6, 7 | eqeq12d 2752 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐿 + 𝑥) = 𝑥 ↔ (𝐿 + 𝑦) = 𝑦)) | 
| 9 | 8 | rspcv 3617 | . . 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 18683 | . . . 4 ⊢ (𝜑 → 𝐿 = 𝑅) | 
| 14 | oveq1 7439 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑅) = (𝑦 + 𝑅)) | |
| 15 | 14, 7 | eqeq12d 2752 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 + 𝑅) = 𝑥 ↔ (𝑦 + 𝑅) = 𝑦)) | 
| 16 | 15 | rspcv 3617 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥 → (𝑦 + 𝑅) = 𝑦)) | 
| 17 | oveq2 7440 | . . . . . . . . 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 2776 | . . . . . . 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 18682 | 1 ⊢ (𝜑 → 𝐿 = 0 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 0gc0g 17485 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-riota 7389 df-ov 7435 df-0g 17487 | 
| This theorem is referenced by: (None) | 
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