Theorem lidrididd 18687

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 18686) 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 7400	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐿 + 𝑥) = (𝐿 + 𝑦))
7		id 22	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
8	6, 7	eqeq12d 2777	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐿 + 𝑥) = 𝑥 ↔ (𝐿 + 𝑦) = 𝑦))
9	8	rspcv 3577	. . 3 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥 → (𝐿 + 𝑦) = 𝑦))
10	5, 9	mpan9 514	. 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 18686	. . . 4 ⊢ (𝜑 → 𝐿 = 𝑅)
14		oveq1 7399	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑅) = (𝑦 + 𝑅))
15	14, 7	eqeq12d 2777	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 + 𝑅) = 𝑥 ↔ (𝑦 + 𝑅) = 𝑦))
16	15	rspcv 3577	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥 → (𝑦 + 𝑅) = 𝑦))
17		oveq2 7400	. . . . . . . . 9 ⊢ (𝐿 = 𝑅 → (𝑦 + 𝐿) = (𝑦 + 𝑅))
18	17	adantl 485	. . . . . . . 8 ⊢ (((𝑦 + 𝑅) = 𝑦 ∧ 𝐿 = 𝑅) → (𝑦 + 𝐿) = (𝑦 + 𝑅))
19		simpl 486	. . . . . . . 8 ⊢ (((𝑦 + 𝑅) = 𝑦 ∧ 𝐿 = 𝑅) → (𝑦 + 𝑅) = 𝑦)
20	18, 19	eqtrd 2796	. . . . . . 7 ⊢ (((𝑦 + 𝑅) = 𝑦 ∧ 𝐿 = 𝑅) → (𝑦 + 𝐿) = 𝑦)
21	20	ex 416	. . . . . 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 410	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 + 𝐿) = 𝑦)
26	1, 2, 3, 4, 10, 25	ismgmid2 18685	1 ⊢ (𝜑 → 𝐿 = 0 )

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  0_gc0g 17451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-riota 7349  df-ov 7395  df-0g 17453

This theorem is referenced by: (None)