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Theorem sgrp0 17900
 Description: Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.)
Assertion
Ref Expression
sgrp0 ((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Smgrp)

Proof of Theorem sgrp0
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgm0 17858 . 2 ((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Mgm)
2 rzal 4451 . . 3 ((Base‘𝑀) = ∅ → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
32adantl 484 . 2 ((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
4 eqid 2819 . . 3 (Base‘𝑀) = (Base‘𝑀)
5 eqid 2819 . . 3 (+g𝑀) = (+g𝑀)
64, 5issgrp 17894 . 2 (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧))))
71, 3, 6sylanbrc 585 1 ((𝑀𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Smgrp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1530   ∈ wcel 2107  ∀wral 3136  ∅c0 4289  ‘cfv 6348  (class class class)co 7148  Basecbs 16475  +gcplusg 16557  Mgmcmgm 17842  Smgrpcsgrp 17892 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-nul 5201 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7151  df-mgm 17844  df-sgrp 17893 This theorem is referenced by: (None)
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