Theorem ismgmd 45370

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 1118	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
3	2	ralrimivva 3191	. . 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 7312	. . . . . 6 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦))
7	6, 4	eleq12d 2828	. . . . 5 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)))
8	4, 7	raleqbidv 3338	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)))
9	4, 8	raleqbidv 3338	. . 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 231	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))
11		ismgmd.0	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
12		eqid 2733	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2733	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
14	12, 13	ismgm 18355	. . 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 256	1 ⊢ (𝜑 → 𝐺 ∈ Mgm)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1537   ∈ wcel 2101  ∀wral 3059  ‘cfv 6447  (class class class)co 7295  Basecbs 16940  +_gcplusg 16990  Mgmcmgm 18352

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 2103  ax-9 2111  ax-ext 2704  ax-nul 5233

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2939  df-ral 3060  df-rab 3224  df-v 3436  df-sbc 3719  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-iota 6399  df-fv 6455  df-ov 7298  df-mgm 18354

This theorem is referenced by:  issubmgm2  45384