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Theorem isgrpi 18206
 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 + = (+g𝐺)
isgrpi.c ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
isgrpi.a ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
isgrpi.z 0𝐵
isgrpi.i (𝑥𝐵 → ( 0 + 𝑥) = 𝑥)
isgrpi.n (𝑥𝐵𝑁𝐵)
isgrpi.j (𝑥𝐵 → (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
StepHypRef Expression
1 isgrpi.b . . . 4 𝐵 = (Base‘𝐺)
21a1i 11 . . 3 (⊤ → 𝐵 = (Base‘𝐺))
3 isgrpi.p . . . 4 + = (+g𝐺)
43a1i 11 . . 3 (⊤ → + = (+g𝐺))
5 isgrpi.c . . . 4 ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
653adant1 1127 . . 3 ((⊤ ∧ 𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
7 isgrpi.a . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
87adantl 485 . . 3 ((⊤ ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9 isgrpi.z . . . 4 0𝐵
109a1i 11 . . 3 (⊤ → 0𝐵)
11 isgrpi.i . . . 4 (𝑥𝐵 → ( 0 + 𝑥) = 𝑥)
1211adantl 485 . . 3 ((⊤ ∧ 𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
13 isgrpi.n . . . 4 (𝑥𝐵𝑁𝐵)
1413adantl 485 . . 3 ((⊤ ∧ 𝑥𝐵) → 𝑁𝐵)
15 isgrpi.j . . . 4 (𝑥𝐵 → (𝑁 + 𝑥) = 0 )
1615adantl 485 . . 3 ((⊤ ∧ 𝑥𝐵) → (𝑁 + 𝑥) = 0 )
172, 4, 6, 8, 10, 12, 14, 16isgrpd 18205 . 2 (⊤ → 𝐺 ∈ Grp)
1817mptru 1545 1 𝐺 ∈ Grp
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  ‘cfv 6340  (class class class)co 7156  Basecbs 16554  +gcplusg 16636  Grpcgrp 18182 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 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6299  df-fun 6342  df-fv 6348  df-riota 7114  df-ov 7159  df-0g 16786  df-mgm 17931  df-sgrp 17980  df-mnd 17991  df-grp 18185 This theorem is referenced by:  isgrpix  18210  cnaddabl  19070  cncrng  20200
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