Theorem ismgmd 41566

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 1258	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
3	2	ralrimivva 2954	. . 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 6566	. . . . . 6 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦))
7	6, 4	eleq12d 2682	. . . . 5 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)))
8	4, 7	raleqbidv 3129	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)))
9	4, 8	raleqbidv 3129	. . 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 221	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
11		ismgmd.0	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
12		eqid 2610	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2610	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
14	12, 13	ismgm 17066	. . 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 246	1 ⊢ (𝜑 → 𝐺 ∈ Mgm)

Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  Mgmcmgm 17063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-mgm 17065

This theorem is referenced by:  issubmgm2  41580