Theorem mgmidmo 13518

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 109	. . . . 5 ⊢ (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → (𝑢 + 𝑥) = 𝑥)
2	1	ralimi 2596	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → ∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥)
3		simpr 110	. . . . 5 ⊢ (((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → (𝑥 + 𝑤) = 𝑥)
4	3	ralimi 2596	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥)
5		oveq1 6035	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 + 𝑤) = (𝑢 + 𝑤))
6		id 19	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
7	5, 6	eqeq12d 2246	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 + 𝑤) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑢))
8	7	rspcva 2909	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥) → (𝑢 + 𝑤) = 𝑢)
9		oveq2 6036	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑢 + 𝑥) = (𝑢 + 𝑤))
10		id 19	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
11	9, 10	eqeq12d 2246	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑤))
12	11	rspcva 2909	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥) → (𝑢 + 𝑤) = 𝑤)
13	8, 12	sylan9req 2285	. . . . . 6 ⊢ (((𝑢 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥) ∧ (𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥)) → 𝑢 = 𝑤)
14	13	an42s 593	. . . . 5 ⊢ (((𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤)
15	14	ex 115	. . . 4 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥) → 𝑢 = 𝑤))
16	2, 4, 15	syl2ani 408	. . 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 2619	. 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 6035	. . . . 5 ⊢ (𝑢 = 𝑤 → (𝑢 + 𝑥) = (𝑤 + 𝑥))
19	18	eqeq1d 2240	. . . 4 ⊢ (𝑢 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑤 + 𝑥) = 𝑥))
20	19	ovanraleqv 6052	. . 3 ⊢ (𝑢 = 𝑤 → (∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)))
21	20	rmo4 3000	. 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 146	1 ⊢ ∃*𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃*wrmo 2514  (class class class)co 6028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rmo 2519  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  ismgmid  13523  mndideu  13572