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

Theorem ismgmd 43920
Description: Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.)
Hypotheses
Ref Expression
ismgmd.b (𝜑𝐵 = (Base‘𝐺))
ismgmd.0 (𝜑𝐺𝑉)
ismgmd.p (𝜑+ = (+g𝐺))
ismgmd.c ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
Assertion
Ref Expression
ismgmd (𝜑𝐺 ∈ Mgm)
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   + (𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ismgmd
StepHypRef Expression
1 ismgmd.c . . . . 5 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
213expb 1112 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
32ralrimivva 3188 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ 𝐵)
4 ismgmd.b . . . 4 (𝜑𝐵 = (Base‘𝐺))
5 ismgmd.p . . . . . . 7 (𝜑+ = (+g𝐺))
65oveqd 7162 . . . . . 6 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
76, 4eleq12d 2904 . . . . 5 (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
84, 7raleqbidv 3399 . . . 4 (𝜑 → (∀𝑦𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
94, 8raleqbidv 3399 . . 3 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
103, 9mpbid 233 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
11 ismgmd.0 . . 3 (𝜑𝐺𝑉)
12 eqid 2818 . . . 4 (Base‘𝐺) = (Base‘𝐺)
13 eqid 2818 . . . 4 (+g𝐺) = (+g𝐺)
1412, 13ismgm 17841 . . 3 (𝐺𝑉 → (𝐺 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
1511, 14syl 17 . 2 (𝜑 → (𝐺 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
1610, 15mpbird 258 1 (𝜑𝐺 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  Mgmcmgm 17838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-mgm 17840
This theorem is referenced by:  issubmgm2  43934
  Copyright terms: Public domain W3C validator