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Theorem ismgmd 44336
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 1117 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
32ralrimivva 3181 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ 𝐵)
4 ismgmd.b . . . 4 (𝜑𝐵 = (Base‘𝐺))
5 ismgmd.p . . . . . . 7 (𝜑+ = (+g𝐺))
65oveqd 7157 . . . . . 6 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
76, 4eleq12d 2908 . . . . 5 (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
84, 7raleqbidv 3382 . . . 4 (𝜑 → (∀𝑦𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
94, 8raleqbidv 3382 . . 3 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
103, 9mpbid 235 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
11 ismgmd.0 . . 3 (𝜑𝐺𝑉)
12 eqid 2822 . . . 4 (Base‘𝐺) = (Base‘𝐺)
13 eqid 2822 . . . 4 (+g𝐺) = (+g𝐺)
1412, 13ismgm 17844 . . 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 260 1 (𝜑𝐺 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2114  wral 3130  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  Mgmcmgm 17841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-mgm 17843
This theorem is referenced by:  issubmgm2  44350
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