Theorem isgrpd 18117

Description: Deduce a group from its properties. Unlike isgrpd2 18115, this one goes straight from the base properties rather than going through Mnd. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
isgrpd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
isgrpd.p	⊢ (𝜑 → + = (+g‘𝐺))
isgrpd.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
isgrpd.a	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
isgrpd.z	⊢ (𝜑 → 0 ∈ 𝐵)
isgrpd.i	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
isgrpd.n	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)
isgrpd.j	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )

Assertion

Ref	Expression
isgrpd	⊢ (𝜑 → 𝐺 ∈ Grp)

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

Allowed substitution hints:   𝑁(𝑥,𝑧)

Proof of Theorem isgrpd

Step	Hyp	Ref	Expression
1		isgrpd.b	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
2		isgrpd.p	. 2 ⊢ (𝜑 → + = (+g‘𝐺))
3		isgrpd.c	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
4		isgrpd.a	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
5		isgrpd.z	. 2 ⊢ (𝜑 → 0 ∈ 𝐵)
6		isgrpd.i	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
7		isgrpd.n	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)
8		isgrpd.j	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )
9		oveq1 7142	. . . . 5 ⊢ (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥))
10	9	eqeq1d 2800	. . . 4 ⊢ (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 ))
11	10	rspcev 3571	. . 3 ⊢ ((𝑁 ∈ 𝐵 ∧ (𝑁 + 𝑥) = 0 ) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
12	7, 8, 11	syl2anc 587	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
13	1, 2, 3, 4, 5, 6, 12	isgrpde 18116	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  Grpcgrp 18095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-riota 7093  df-ov 7138  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098

This theorem is referenced by:  isgrpi  18118  issubg2  18286  symggrp  18520  isdrngd  19520  psrgrp  20636  cnlmod  23745  dchrabl  25838  motgrp  26337  ldualgrplem  36441  tgrpgrplem  38045  erngdvlem1  38284  erngdvlem1-rN  38292  dvhgrp  38403  mendring  40136