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Theorem mgmidmo 18628
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 481 . . . . 5 (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → (𝑢 + 𝑥) = 𝑥)
21ralimi 3072 . . . 4 (∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → ∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥)
3 simpr 483 . . . . 5 (((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → (𝑥 + 𝑤) = 𝑥)
43ralimi 3072 . . . 4 (∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥)
5 oveq1 7426 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑥 + 𝑤) = (𝑢 + 𝑤))
6 id 22 . . . . . . . . 9 (𝑥 = 𝑢𝑥 = 𝑢)
75, 6eqeq12d 2741 . . . . . . . 8 (𝑥 = 𝑢 → ((𝑥 + 𝑤) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑢))
87rspcva 3604 . . . . . . 7 ((𝑢𝐵 ∧ ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥) → (𝑢 + 𝑤) = 𝑢)
9 oveq2 7427 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑢 + 𝑥) = (𝑢 + 𝑤))
10 id 22 . . . . . . . . 9 (𝑥 = 𝑤𝑥 = 𝑤)
119, 10eqeq12d 2741 . . . . . . . 8 (𝑥 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑤))
1211rspcva 3604 . . . . . . 7 ((𝑤𝐵 ∧ ∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥) → (𝑢 + 𝑤) = 𝑤)
138, 12sylan9req 2786 . . . . . 6 (((𝑢𝐵 ∧ ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥) ∧ (𝑤𝐵 ∧ ∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥)) → 𝑢 = 𝑤)
1413an42s 659 . . . . 5 (((𝑢𝐵𝑤𝐵) ∧ (∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤)
1514ex 411 . . . 4 ((𝑢𝐵𝑤𝐵) → ((∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥) → 𝑢 = 𝑤))
162, 4, 15syl2ani 605 . . 3 ((𝑢𝐵𝑤𝐵) → ((∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤))
1716rgen2 3187 . 2 𝑢𝐵𝑤𝐵 ((∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤)
18 oveq1 7426 . . . . 5 (𝑢 = 𝑤 → (𝑢 + 𝑥) = (𝑤 + 𝑥))
1918eqeq1d 2727 . . . 4 (𝑢 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑤 + 𝑥) = 𝑥))
2019ovanraleqv 7443 . . 3 (𝑢 = 𝑤 → (∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ ∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)))
2120rmo4 3722 . 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))
2217, 21mpbir 230 1 ∃*𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3050  ∃*wrmo 3362  (class class class)co 7419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rmo 3363  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422
This theorem is referenced by:  ismgmid  18633  mndideu  18713
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