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Theorem isgrpi 17758
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 1161 . . 3 ((⊤ ∧ 𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
7 isgrpi.a . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
87adantl 474 . . 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 474 . . 3 ((⊤ ∧ 𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
13 isgrpi.n . . . 4 (𝑥𝐵𝑁𝐵)
1413adantl 474 . . 3 ((⊤ ∧ 𝑥𝐵) → 𝑁𝐵)
15 isgrpi.j . . . 4 (𝑥𝐵 → (𝑁 + 𝑥) = 0 )
1615adantl 474 . . 3 ((⊤ ∧ 𝑥𝐵) → (𝑁 + 𝑥) = 0 )
172, 4, 6, 8, 10, 12, 14, 16isgrpd 17757 . 2 (⊤ → 𝐺 ∈ Grp)
1817mptru 1661 1 𝐺 ∈ Grp
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wtru 1654  wcel 2157  cfv 6100  (class class class)co 6877  Basecbs 16181  +gcplusg 16264  Grpcgrp 17735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3386  df-sbc 3633  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6063  df-fun 6102  df-fv 6108  df-riota 6838  df-ov 6880  df-0g 16414  df-mgm 17554  df-sgrp 17596  df-mnd 17607  df-grp 17738
This theorem is referenced by:  isgrpix  17762  cnaddabl  18584  cncrng  20086
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