Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  0mgm Structured version   Visualization version   GIF version

Theorem 0mgm 48082
Description: A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.)
Hypothesis
Ref Expression
0mgm.b (Base‘𝑀) = ∅
Assertion
Ref Expression
0mgm (𝑀𝑉𝑀 ∈ Mgm)

Proof of Theorem 0mgm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 4513 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g𝑀)𝑦) ∈ ∅
2 0mgm.b . . . 4 (Base‘𝑀) = ∅
32eqcomi 2746 . . 3 ∅ = (Base‘𝑀)
4 eqid 2737 . . 3 (+g𝑀) = (+g𝑀)
53, 4ismgm 18654 . 2 (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g𝑀)𝑦) ∈ ∅))
61, 5mpbiri 258 1 (𝑀𝑉𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  c0 4333  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Mgmcmgm 18651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-mgm 18653
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator