Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  isgrpi Structured version   Visualization version   GIF version

Theorem isgrpi 17652
 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 1123 . . 3 ((⊤ ∧ 𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
7 isgrpi.a . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
87adantl 467 . . 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 467 . . 3 ((⊤ ∧ 𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
13 isgrpi.n . . . 4 (𝑥𝐵𝑁𝐵)
1413adantl 467 . . 3 ((⊤ ∧ 𝑥𝐵) → 𝑁𝐵)
15 isgrpi.j . . . 4 (𝑥𝐵 → (𝑁 + 𝑥) = 0 )
1615adantl 467 . . 3 ((⊤ ∧ 𝑥𝐵) → (𝑁 + 𝑥) = 0 )
172, 4, 6, 8, 10, 12, 14, 16isgrpd 17651 . 2 (⊤ → 𝐺 ∈ Grp)
1817trud 1640 1 𝐺 ∈ Grp
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1070   = wceq 1630  ⊤wtru 1631   ∈ wcel 2144  ‘cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  Grpcgrp 17629 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-riota 6753  df-ov 6795  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632 This theorem is referenced by:  isgrpix  17656  cnaddabl  18478  cncrng  19981
 Copyright terms: Public domain W3C validator