Theorem isgrpd2e 18936

Description: Deduce a group from its properties. In this version of isgrpd2 18937, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 10-Aug-2013.)

Hypotheses

Ref	Expression
isgrpd2.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
isgrpd2.p	⊢ (𝜑 → + = (+g‘𝐺))
isgrpd2.z	⊢ (𝜑 → 0 = (0g‘𝐺))
isgrpd2.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
isgrpd2e.n	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )

Assertion

Ref	Expression
isgrpd2e	⊢ (𝜑 → 𝐺 ∈ Grp)

Distinct variable groups:   𝑥,𝑦, +   𝑦, 0   𝑥,𝐵,𝑦   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦

Allowed substitution hint:   0 (𝑥)

Proof of Theorem isgrpd2e

Step	Hyp	Ref	Expression
1		isgrpd2.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
2		isgrpd2e.n	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
3	2	ralrimiva 3132	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
4		isgrpd2.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
5		isgrpd2.p	. . . . . . 7 ⊢ (𝜑 → + = (+g‘𝐺))
6	5	oveqd 7420	. . . . . 6 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐺)𝑥))
7		isgrpd2.z	. . . . . 6 ⊢ (𝜑 → 0 = (0g‘𝐺))
8	6, 7	eqeq12d 2751	. . . . 5 ⊢ (𝜑 → ((𝑦 + 𝑥) = 0 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
9	4, 8	rexeqbidv 3326	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
10	4, 9	raleqbidv 3325	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ↔ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
11	3, 10	mpbid 232	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))
12		eqid 2735	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2735	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
14		eqid 2735	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
15	12, 13, 14	isgrp 18920	. 2 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
16	1, 11, 15	sylanbrc 583	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  ‘cfv 6530  (class class class)co 7403  Basecbs 17226  +_gcplusg 17269  0_gc0g 17451  Mndcmnd 18710  Grpcgrp 18914

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6483  df-fv 6538  df-ov 7406  df-grp 18917

This theorem is referenced by:  isgrpd2  18937  isgrpde  18938  rloccring  33211