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Theorem ismgmd 18709

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 1136	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
3	2	ralrimivva 3214	. . 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 7428	. . . . . 6 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+_g‘𝐺)𝑦))
7	6, 4	eleq12d 2863	. . . . 5 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺)))
8	4, 7	raleqbidv 3345	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺)))
9	4, 8	raleqbidv 3345	. . 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 235	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺))
11		ismgmd.0	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
12		eqid 2769	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2769	. . . 4 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
14	12, 13	ismgm 18698	. . 3 ⊢ (𝐺 ∈ 𝑉 → (𝐺 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺)))
15	11, 14	syl 18	. 2 ⊢ (𝜑 → (𝐺 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+_g‘𝐺)𝑦) ∈ (Base‘𝐺)))
16	10, 15	mpbird 260	1 ⊢ (𝜑 → 𝐺 ∈ Mgm)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ‘cfv 6537 (class class class)co 7411 Basecbs 17268 +_gcplusg 17309 Mgmcmgm 18695

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-ext 2741 ax-nul 5271

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-sb 2098 df-clab 2748 df-cleq 2761 df-clel 2844 df-ne 2965 df-ral 3086 df-rab 3424 df-v 3465 df-sbc 3754 df-dif 3916 df-un 3918 df-ss 3930 df-nul 4295 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-iota 6493 df-fv 6545 df-ov 7414 df-mgm 18697

This theorem is referenced by: issubmgm2 18760

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