Theorem isgrpd2 18842

Description: Deduce a group from its properties. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). Note: normally we don't use a 𝜑 antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2733, but we make an exception for theorems such as isgrpd2 18842, ismndd 18647, and islmodd 20477 since theorems using them often rewrite the structure components. (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)
isgrpd2.n	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)
isgrpd2.j	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )

Assertion

Ref	Expression
isgrpd2	⊢ (𝜑 → 𝐺 ∈ Grp)

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

Allowed substitution hints:   𝑁(𝑥)   0 (𝑥)

Proof of Theorem isgrpd2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isgrpd2.b	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
2		isgrpd2.p	. 2 ⊢ (𝜑 → + = (+g‘𝐺))
3		isgrpd2.z	. 2 ⊢ (𝜑 → 0 = (0g‘𝐺))
4		isgrpd2.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
5		isgrpd2.n	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)
6		isgrpd2.j	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )
7		oveq1 7416	. . . . 5 ⊢ (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥))
8	7	eqeq1d 2735	. . . 4 ⊢ (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 ))
9	8	rspcev 3613	. . 3 ⊢ ((𝑁 ∈ 𝐵 ∧ (𝑁 + 𝑥) = 0 ) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
10	5, 6, 9	syl2anc 585	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
11	1, 2, 3, 4, 10	isgrpd2e 18841	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  +_gcplusg 17197  0_gc0g 17385  Mndcmnd 18625  Grpcgrp 18819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-grp 18822

This theorem is referenced by:  prdsgrpd  18933  oppggrp  19224