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Theorem ismgmd 45861
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 1121 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
32ralrimivva 3196 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ 𝐵)
4 ismgmd.b . . . 4 (𝜑𝐵 = (Base‘𝐺))
5 ismgmd.p . . . . . . 7 (𝜑+ = (+g𝐺))
65oveqd 7367 . . . . . 6 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
76, 4eleq12d 2833 . . . . 5 (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
84, 7raleqbidv 3318 . . . 4 (𝜑 → (∀𝑦𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
94, 8raleqbidv 3318 . . 3 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
103, 9mpbid 231 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
11 ismgmd.0 . . 3 (𝜑𝐺𝑉)
12 eqid 2738 . . . 4 (Base‘𝐺) = (Base‘𝐺)
13 eqid 2738 . . . 4 (+g𝐺) = (+g𝐺)
1412, 13ismgm 18434 . . 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 205  w3a 1088   = wceq 1542  wcel 2107  wral 3063  cfv 6492  (class class class)co 7350  Basecbs 17019  +gcplusg 17069  Mgmcmgm 18431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-nul 5262
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-ral 3064  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-iota 6444  df-fv 6500  df-ov 7353  df-mgm 18433
This theorem is referenced by:  issubmgm2  45875
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