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Theorem ismgmd 18586

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 3181	. . 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 7407	. . . . . 6 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+_g‘𝐺)𝑦))
7	6, 4	eleq12d 2823	. . . . 5 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺)))
8	4, 7	raleqbidv 3321	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺)))
9	4, 8	raleqbidv 3321	. . 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 2730	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2730	. . . 4 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
14	12, 13	ismgm 18575	. . 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 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 +_gcplusg 17227 Mgmcmgm 18572

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2702 ax-nul 5264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2709 df-cleq 2722 df-clel 2804 df-ne 2927 df-ral 3046 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-iota 6467 df-fv 6522 df-ov 7393 df-mgm 18574

This theorem is referenced by: issubmgm2 18637

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