Theorem isgrpi 18602

Description: Properties that determine a group. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). (Contributed by NM, 3-Sep-2011.)

Hypotheses

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

Assertion

Ref	Expression
isgrpi	⊢ 𝐺 ∈ Grp

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

Allowed substitution hints:   𝑁(𝑥,𝑧)

Proof of Theorem isgrpi

Step	Hyp	Ref	Expression
1		isgrpi.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2	1	a1i 11	. . 3 ⊢ (⊤ → 𝐵 = (Base‘𝐺))
3		isgrpi.p	. . . 4 ⊢ + = (+g‘𝐺)
4	3	a1i 11	. . 3 ⊢ (⊤ → + = (+g‘𝐺))
5		isgrpi.c	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
6	5	3adant1 1129	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
7		isgrpi.a	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
8	7	adantl 482	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9		isgrpi.z	. . . 4 ⊢ 0 ∈ 𝐵
10	9	a1i 11	. . 3 ⊢ (⊤ → 0 ∈ 𝐵)
11		isgrpi.i	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ( 0 + 𝑥) = 𝑥)
12	11	adantl 482	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
13		isgrpi.n	. . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑁 ∈ 𝐵)
14	13	adantl 482	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)
15		isgrpi.j	. . . 4 ⊢ (𝑥 ∈ 𝐵 → (𝑁 + 𝑥) = 0 )
16	15	adantl 482	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )
17	2, 4, 6, 8, 10, 12, 14, 16	isgrpd 18601	. 2 ⊢ (⊤ → 𝐺 ∈ Grp)
18	17	mptru 1546	1 ⊢ 𝐺 ∈ Grp

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ⊤wtru 1540   ∈ wcel 2106  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  Grpcgrp 18577

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-riota 7232  df-ov 7278  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580

This theorem is referenced by:  isgrpix  18606  cnaddabl  19470  cncrng  20619