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 41559
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 4027 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g𝑀)𝑦) ∈ ∅
2 0mgm.b . . . 4 (Base‘𝑀) = ∅
32eqcomi 2618 . . 3 ∅ = (Base‘𝑀)
4 eqid 2609 . . 3 (+g𝑀) = (+g𝑀)
53, 4ismgm 17012 . 2 (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g𝑀)𝑦) ∈ ∅))
61, 5mpbiri 246 1 (𝑀𝑉𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wral 2895  c0 3873  cfv 5790  (class class class)co 6527  Basecbs 15641  +gcplusg 15714  Mgmcmgm 17009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ov 6530  df-mgm 17011
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator