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Theorem mgmidmo 16978
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 ∃*𝑢𝐵𝑥𝐵 ((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.
StepHypRef Expression
1 simpl 471 . . . . 5 (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → (𝑢 + 𝑥) = 𝑥)
21ralimi 2840 . . . 4 (∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → ∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥)
3 simpr 475 . . . . 5 (((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → (𝑥 + 𝑤) = 𝑥)
43ralimi 2840 . . . 4 (∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥)
5 oveq1 6432 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑥 + 𝑤) = (𝑢 + 𝑤))
6 id 22 . . . . . . . . 9 (𝑥 = 𝑢𝑥 = 𝑢)
75, 6eqeq12d 2529 . . . . . . . 8 (𝑥 = 𝑢 → ((𝑥 + 𝑤) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑢))
87rspcva 3184 . . . . . . 7 ((𝑢𝐵 ∧ ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥) → (𝑢 + 𝑤) = 𝑢)
9 oveq2 6433 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑢 + 𝑥) = (𝑢 + 𝑤))
10 id 22 . . . . . . . . 9 (𝑥 = 𝑤𝑥 = 𝑤)
119, 10eqeq12d 2529 . . . . . . . 8 (𝑥 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑤))
1211rspcva 3184 . . . . . . 7 ((𝑤𝐵 ∧ ∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥) → (𝑢 + 𝑤) = 𝑤)
138, 12sylan9req 2569 . . . . . 6 (((𝑢𝐵 ∧ ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥) ∧ (𝑤𝐵 ∧ ∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥)) → 𝑢 = 𝑤)
1413an42s 865 . . . . 5 (((𝑢𝐵𝑤𝐵) ∧ (∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤)
1514ex 448 . . . 4 ((𝑢𝐵𝑤𝐵) → ((∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥) → 𝑢 = 𝑤))
162, 4, 15syl2ani 685 . . 3 ((𝑢𝐵𝑤𝐵) → ((∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤))
1716rgen2a 2864 . 2 𝑢𝐵𝑤𝐵 ((∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤)
18 oveq1 6432 . . . . . 6 (𝑢 = 𝑤 → (𝑢 + 𝑥) = (𝑤 + 𝑥))
1918eqeq1d 2516 . . . . 5 (𝑢 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑤 + 𝑥) = 𝑥))
20 oveq2 6433 . . . . . 6 (𝑢 = 𝑤 → (𝑥 + 𝑢) = (𝑥 + 𝑤))
2120eqeq1d 2516 . . . . 5 (𝑢 = 𝑤 → ((𝑥 + 𝑢) = 𝑥 ↔ (𝑥 + 𝑤) = 𝑥))
2219, 21anbi12d 742 . . . 4 (𝑢 = 𝑤 → (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)))
2322ralbidv 2873 . . 3 (𝑢 = 𝑤 → (∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ ∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)))
2423rmo4 3270 . 2 (∃*𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ ∀𝑢𝐵𝑤𝐵 ((∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤))
2517, 24mpbir 219 1 ∃*𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  wral 2800  ∃*wrmo 2803  (class class class)co 6425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rmo 2808  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-iota 5653  df-fv 5697  df-ov 6428
This theorem is referenced by:  ismgmid  16983  mndideu  17023
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