MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismgmd Structured version   Visualization version   GIF version

Theorem ismgmd 18678
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 1119 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
32ralrimivva 3200 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ 𝐵)
4 ismgmd.b . . . 4 (𝜑𝐵 = (Base‘𝐺))
5 ismgmd.p . . . . . . 7 (𝜑+ = (+g𝐺))
65oveqd 7448 . . . . . 6 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
76, 4eleq12d 2833 . . . . 5 (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
84, 7raleqbidv 3344 . . . 4 (𝜑 → (∀𝑦𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
94, 8raleqbidv 3344 . . 3 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
103, 9mpbid 232 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
11 ismgmd.0 . . 3 (𝜑𝐺𝑉)
12 eqid 2735 . . . 4 (Base‘𝐺) = (Base‘𝐺)
13 eqid 2735 . . . 4 (+g𝐺) = (+g𝐺)
1412, 13ismgm 18667 . . 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 257 1 (𝜑𝐺 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Mgmcmgm 18664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-mgm 18666
This theorem is referenced by:  issubmgm2  18729
  Copyright terms: Public domain W3C validator