Theorem mgmidmo 18717

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 487	. . . . 5 ⊢ (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → (𝑢 + 𝑥) = 𝑥)
2	1	ralimi 3108	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → ∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥)
3		simpr 489	. . . . 5 ⊢ (((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → (𝑥 + 𝑤) = 𝑥)
4	3	ralimi 3108	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥)
5		oveq1 7418	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 + 𝑤) = (𝑢 + 𝑤))
6		id 23	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
7	5, 6	eqeq12d 2785	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 + 𝑤) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑢))
8	7	rspcva 3588	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥) → (𝑢 + 𝑤) = 𝑢)
9		oveq2 7419	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑢 + 𝑥) = (𝑢 + 𝑤))
10		id 23	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
11	9, 10	eqeq12d 2785	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑤))
12	11	rspcva 3588	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥) → (𝑢 + 𝑤) = 𝑤)
13	8, 12	sylan9req 2825	. . . . . 6 ⊢ (((𝑢 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥) ∧ (𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥)) → 𝑢 = 𝑤)
14	13	an42s 673	. . . . 5 ⊢ (((𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤)
15	14	ex 417	. . . 4 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥) → 𝑢 = 𝑤))
16	2, 4, 15	syl2ani 618	. . 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 3211	. 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 7418	. . . . 5 ⊢ (𝑢 = 𝑤 → (𝑢 + 𝑥) = (𝑤 + 𝑥))
19	18	eqeq1d 2771	. . . 4 ⊢ (𝑢 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑤 + 𝑥) = 𝑥))
20	19	ovanraleqv 7435	. . 3 ⊢ (𝑢 = 𝑤 → (∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)))
21	20	rmo4 3702	. 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)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃*wrmo 3375  (class class class)co 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rmo 3376  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414

This theorem is referenced by:  ismgmid  18722  mndideu  18802