Theorem mgmidmo 18742

Description: A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by NM, 17-Jun-2017.)

Assertion

Ref	Expression
mgmidmo	⊢ ∃*𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)

Distinct variable groups:   𝑥,𝑢,𝐵   𝑢, + ,𝑥

Proof of Theorem mgmidmo

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 488	. . . . 5 ⊢ (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → (𝑢 + 𝑥) = 𝑥)
2	1	ralimi 3104	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → ∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥)
3		simpr 490	. . . . 5 ⊢ (((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → (𝑥 + 𝑤) = 𝑥)
4	3	ralimi 3104	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥)
5		oveq1 7426	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 + 𝑤) = (𝑢 + 𝑤))
6		id 23	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
7	5, 6	eqeq12d 2781	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 + 𝑤) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑢))
8	7	rspcva 3581	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥) → (𝑢 + 𝑤) = 𝑢)
9		oveq2 7427	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑢 + 𝑥) = (𝑢 + 𝑤))
10		id 23	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
11	9, 10	eqeq12d 2781	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑤))
12	11	rspcva 3581	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥) → (𝑢 + 𝑤) = 𝑤)
13	8, 12	sylan9req 2821	. . . . . 6 ⊢ (((𝑢 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥) ∧ (𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥)) → 𝑢 = 𝑤)
14	13	an42s 674	. . . . 5 ⊢ (((𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤)
15	14	ex 418	. . . 4 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥) → 𝑢 = 𝑤))
16	2, 4, 15	syl2ani 619	. . 3 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤))
17	16	rgen2 3207	. 2 ⊢ ∀𝑢 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤)
18		oveq1 7426	. . . . 5 ⊢ (𝑢 = 𝑤 → (𝑢 + 𝑥) = (𝑤 + 𝑥))
19	18	eqeq1d 2767	. . . 4 ⊢ (𝑢 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑤 + 𝑥) = 𝑥))
20	19	ovanraleqv 7443	. . 3 ⊢ (𝑢 = 𝑤 → (∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)))
21	20	rmo4 3695	. 2 ⊢ (∃*𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ ∀𝑢 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ((∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∀𝑥 ∈ 𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤))
22	17, 21	mpbir 234	1 ⊢ ∃*𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃*wrmo 3370  (class class class)co 7419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rmo 3371  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422

This theorem is used by:  mgmideud  18743  ismgmid  18748  mndideuOLD  18836