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Theorem ismgmd 18690

Description: Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.)

**Hypotheses**
Ref	Expression
ismgmd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
ismgmd.0	⊢ (𝜑 → 𝐺 ∈ 𝑉)
ismgmd.p	⊢ (𝜑 → + = (+_g‘𝐺))
ismgmd.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)

**Assertion**
Ref	Expression
ismgmd	⊢ (𝜑 → 𝐺 ∈ 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**
Step	Hyp	Ref	Expression
1		ismgmd.c	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
2	1	3expb 1120	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
3	2	ralrimivva 3208	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵)
4		ismgmd.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
5		ismgmd.p	. . . . . . 7 ⊢ (𝜑 → + = (+_g‘𝐺))
6	5	oveqd 7465	. . . . . 6 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+_g‘𝐺)𝑦))
7	6, 4	eleq12d 2838	. . . . 5 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺)))
8	4, 7	raleqbidv 3354	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺)))
9	4, 8	raleqbidv 3354	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺)))
10	3, 9	mpbid 232	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺))
11		ismgmd.0	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
12		eqid 2740	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2740	. . . 4 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
14	12, 13	ismgm 18679	. . 3 ⊢ (𝐺 ∈ 𝑉 → (𝐺 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺)))
15	11, 14	syl 17	. 2 ⊢ (𝜑 → (𝐺 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺)))
16	10, 15	mpbird 257	1 ⊢ (𝜑 → 𝐺 ∈ Mgm)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 +_gcplusg 17311 Mgmcmgm 18676

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-nul 5324

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ne 2947 df-ral 3068 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-iota 6525 df-fv 6581 df-ov 7451 df-mgm 18678

This theorem is referenced by: issubmgm2 18741

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