Theorem isgrpd2e 18124

Description: Deduce a group from its properties. In this version of isgrpd2 18125, 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 3184	. . 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 7175	. . . . . 6 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐺)𝑥))
7		isgrpd2.z	. . . . . 6 ⊢ (𝜑 → 0 = (0g‘𝐺))
8	6, 7	eqeq12d 2839	. . . . 5 ⊢ (𝜑 → ((𝑦 + 𝑥) = 0 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
9	4, 8	rexeqbidv 3404	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
10	4, 9	raleqbidv 3403	. . 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 234	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))
12		eqid 2823	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2823	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
14		eqid 2823	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
15	12, 13, 14	isgrp 18111	. 2 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
16	1, 11, 15	sylanbrc 585	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  +_gcplusg 16567  0_gc0g 16715  Mndcmnd 17913  Grpcgrp 18105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161  df-grp 18108

This theorem is referenced by:  isgrpd2  18125  isgrpde  18126