Theorem isgrpi 18889

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 1130	. . 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 481	. . 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 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
13		isgrpi.n	. . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑁 ∈ 𝐵)
14	13	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)
15		isgrpi.j	. . . 4 ⊢ (𝑥 ∈ 𝐵 → (𝑁 + 𝑥) = 0 )
16	15	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )
17	2, 4, 6, 8, 10, 12, 14, 16	isgrpd 18888	. 2 ⊢ (⊤ → 𝐺 ∈ Grp)
18	17	mptru 1548	1 ⊢ 𝐺 ∈ Grp

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ⊤wtru 1542   ∈ wcel 2113  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  +_gcplusg 17177  Grpcgrp 18863

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-riota 7315  df-ov 7361  df-0g 17361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866

This theorem is referenced by:  isgrpix  18894  cnaddabl  19798  cncrng  21343  cncrngOLD  21344  zsoring  28405