Theorem mgmidmo 12603

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 108	. . . . 5 ⊢ (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → (𝑢 + 𝑥) = 𝑥)
2	1	ralimi 2529	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → ∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥)
3		simpr 109	. . . . 5 ⊢ (((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → (𝑥 + 𝑤) = 𝑥)
4	3	ralimi 2529	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥)
5		oveq1 5849	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 + 𝑤) = (𝑢 + 𝑤))
6		id 19	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
7	5, 6	eqeq12d 2180	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 + 𝑤) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑢))
8	7	rspcva 2828	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥) → (𝑢 + 𝑤) = 𝑢)
9		oveq2 5850	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑢 + 𝑥) = (𝑢 + 𝑤))
10		id 19	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
11	9, 10	eqeq12d 2180	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑤))
12	11	rspcva 2828	. . . . . . 7 ⊢ ((𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥) → (𝑢 + 𝑤) = 𝑤)
13	8, 12	sylan9req 2220	. . . . . 6 ⊢ (((𝑢 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥) ∧ (𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥)) → 𝑢 = 𝑤)
14	13	an42s 579	. . . . 5 ⊢ (((𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤)
15	14	ex 114	. . . 4 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((∀𝑥 ∈ 𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝐵 (𝑥 + 𝑤) = 𝑥) → 𝑢 = 𝑤))
16	2, 4, 15	syl2ani 406	. . 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 2552	. 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 5849	. . . . 5 ⊢ (𝑢 = 𝑤 → (𝑢 + 𝑥) = (𝑤 + 𝑥))
19	18	eqeq1d 2174	. . . 4 ⊢ (𝑢 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑤 + 𝑥) = 𝑥))
20	19	ovanraleqv 5866	. . 3 ⊢ (𝑢 = 𝑤 → (∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)))
21	20	rmo4 2919	. 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 145	1 ⊢ ∃*𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ∃*wrmo 2447  (class class class)co 5842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rmo 2452  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  ismgmid  12608