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Theorem mgmidmo 17199
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 473 . . . . 5 (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → (𝑢 + 𝑥) = 𝑥)
21ralimi 2948 . . . 4 (∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) → ∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥)
3 simpr 477 . . . . 5 (((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → (𝑥 + 𝑤) = 𝑥)
43ralimi 2948 . . . 4 (∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥) → ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥)
5 oveq1 6622 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑥 + 𝑤) = (𝑢 + 𝑤))
6 id 22 . . . . . . . . 9 (𝑥 = 𝑢𝑥 = 𝑢)
75, 6eqeq12d 2636 . . . . . . . 8 (𝑥 = 𝑢 → ((𝑥 + 𝑤) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑢))
87rspcva 3297 . . . . . . 7 ((𝑢𝐵 ∧ ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥) → (𝑢 + 𝑤) = 𝑢)
9 oveq2 6623 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑢 + 𝑥) = (𝑢 + 𝑤))
10 id 22 . . . . . . . . 9 (𝑥 = 𝑤𝑥 = 𝑤)
119, 10eqeq12d 2636 . . . . . . . 8 (𝑥 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑢 + 𝑤) = 𝑤))
1211rspcva 3297 . . . . . . 7 ((𝑤𝐵 ∧ ∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥) → (𝑢 + 𝑤) = 𝑤)
138, 12sylan9req 2676 . . . . . 6 (((𝑢𝐵 ∧ ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥) ∧ (𝑤𝐵 ∧ ∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥)) → 𝑢 = 𝑤)
1413an42s 869 . . . . 5 (((𝑢𝐵𝑤𝐵) ∧ (∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤)
1514ex 450 . . . 4 ((𝑢𝐵𝑤𝐵) → ((∀𝑥𝐵 (𝑢 + 𝑥) = 𝑥 ∧ ∀𝑥𝐵 (𝑥 + 𝑤) = 𝑥) → 𝑢 = 𝑤))
162, 4, 15syl2ani 687 . . 3 ((𝑢𝐵𝑤𝐵) → ((∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤))
1716rgen2a 2973 . 2 𝑢𝐵𝑤𝐵 ((∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)) → 𝑢 = 𝑤)
18 oveq1 6622 . . . . . 6 (𝑢 = 𝑤 → (𝑢 + 𝑥) = (𝑤 + 𝑥))
1918eqeq1d 2623 . . . . 5 (𝑢 = 𝑤 → ((𝑢 + 𝑥) = 𝑥 ↔ (𝑤 + 𝑥) = 𝑥))
20 oveq2 6623 . . . . . 6 (𝑢 = 𝑤 → (𝑥 + 𝑢) = (𝑥 + 𝑤))
2120eqeq1d 2623 . . . . 5 (𝑢 = 𝑤 → ((𝑥 + 𝑢) = 𝑥 ↔ (𝑥 + 𝑤) = 𝑥))
2219, 21anbi12d 746 . . . 4 (𝑢 = 𝑤 → (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)))
2322ralbidv 2982 . . 3 (𝑢 = 𝑤 → (∀𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ ∀𝑥𝐵 ((𝑤 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑤) = 𝑥)))
2423rmo4 3386 . 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 221 1 ∃*𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2908  ∃*wrmo 2911  (class class class)co 6615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rmo 2916  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618
This theorem is referenced by:  ismgmid  17204  mndideu  17244
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