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Theorem mgm0 18471
Description: Any set with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.)
Assertion
Ref Expression
mgm0 ((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Mgm)

Proof of Theorem mgm0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rzal 4464 . . 3 ((Base‘𝑀) = ∅ → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
21adantl 482 . 2 ((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
3 eqid 2737 . . . 4 (Base‘𝑀) = (Base‘𝑀)
4 eqid 2737 . . . 4 (+g𝑀) = (+g𝑀)
53, 4ismgm 18458 . . 3 (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
65adantr 481 . 2 ((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
72, 6mpbird 256 1 ((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3062  c0 4280  cfv 6493  (class class class)co 7351  Basecbs 17043  +gcplusg 17093  Mgmcmgm 18455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-nul 5261
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-iota 6445  df-fv 6501  df-ov 7354  df-mgm 18457
This theorem is referenced by:  mgm0b  18472  sgrp0  18513
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