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Theorem mgm0 17608
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 4295 . . 3 ((Base‘𝑀) = ∅ → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
21adantl 475 . 2 ((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
3 eqid 2825 . . . 4 (Base‘𝑀) = (Base‘𝑀)
4 eqid 2825 . . . 4 (+g𝑀) = (+g𝑀)
53, 4ismgm 17596 . . 3 (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
65adantr 474 . 2 ((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
72, 6mpbird 249 1 ((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wral 3117  c0 4144  cfv 6123  (class class class)co 6905  Basecbs 16222  +gcplusg 16305  Mgmcmgm 17593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-nul 5013
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-mgm 17595
This theorem is referenced by:  mgm0b  17609  sgrp0  17644
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