Theorem lidrididd 18708

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 18707) is equal to the two-sided identity element. (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)
lidrideqd.b	⊢ 𝐵 = (Base‘𝐺)
lidrideqd.p	⊢ + = (+g‘𝐺)
lidrididd.o	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
lidrididd	⊢ (𝜑 → 𝐿 = 0 )

Distinct variable groups:   𝑥,𝐵   𝑥,𝐿   𝑥,𝑅   𝑥, +

Allowed substitution hints:   𝜑(𝑥)   𝐺(𝑥)   0 (𝑥)

Proof of Theorem lidrididd

Dummy variable 𝑦 is distinct from all other variables.

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 7456	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐿 + 𝑥) = (𝐿 + 𝑦))
7		id 22	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
8	6, 7	eqeq12d 2756	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐿 + 𝑥) = 𝑥 ↔ (𝐿 + 𝑦) = 𝑦))
9	8	rspcv 3631	. . 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 18707	. . . 4 ⊢ (𝜑 → 𝐿 = 𝑅)
14		oveq1 7455	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑅) = (𝑦 + 𝑅))
15	14, 7	eqeq12d 2756	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 + 𝑅) = 𝑥 ↔ (𝑦 + 𝑅) = 𝑦))
16	15	rspcv 3631	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥 → (𝑦 + 𝑅) = 𝑦))
17		oveq2 7456	. . . . . . . . 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 2780	. . . . . . 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 18706	1 ⊢ (𝜑 → 𝐿 = 0 )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  0_gc0g 17499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-riota 7404  df-ov 7451  df-0g 17501

This theorem is referenced by: (None)