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

Theorem isgrpd 17799
Description: Deduce a group from its properties. Unlike isgrpd2 17797, this one goes straight from the base properties rather than going through Mnd. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
isgrpd.b (𝜑𝐵 = (Base‘𝐺))
isgrpd.p (𝜑+ = (+g𝐺))
isgrpd.c ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
isgrpd.a ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
isgrpd.z (𝜑0𝐵)
isgrpd.i ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
isgrpd.n ((𝜑𝑥𝐵) → 𝑁𝐵)
isgrpd.j ((𝜑𝑥𝐵) → (𝑁 + 𝑥) = 0 )
Assertion
Ref Expression
isgrpd (𝜑𝐺 ∈ Grp)
Distinct variable groups:   𝑥,𝑦,𝑧, +   𝑥, 0 ,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑦,𝑁   𝜑,𝑥,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧
Allowed substitution hints:   𝑁(𝑥,𝑧)

Proof of Theorem isgrpd
StepHypRef Expression
1 isgrpd.b . 2 (𝜑𝐵 = (Base‘𝐺))
2 isgrpd.p . 2 (𝜑+ = (+g𝐺))
3 isgrpd.c . 2 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
4 isgrpd.a . 2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
5 isgrpd.z . 2 (𝜑0𝐵)
6 isgrpd.i . 2 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
7 isgrpd.n . . 3 ((𝜑𝑥𝐵) → 𝑁𝐵)
8 isgrpd.j . . 3 ((𝜑𝑥𝐵) → (𝑁 + 𝑥) = 0 )
9 oveq1 6913 . . . . 5 (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥))
109eqeq1d 2828 . . . 4 (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 ))
1110rspcev 3527 . . 3 ((𝑁𝐵 ∧ (𝑁 + 𝑥) = 0 ) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
127, 8, 11syl2anc 581 . 2 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
131, 2, 3, 4, 5, 6, 12isgrpde 17798 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  wrex 3119  cfv 6124  (class class class)co 6906  Basecbs 16223  +gcplusg 16306  Grpcgrp 17777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-iota 6087  df-fun 6126  df-fv 6132  df-riota 6867  df-ov 6909  df-0g 16456  df-mgm 17596  df-sgrp 17638  df-mnd 17649  df-grp 17780
This theorem is referenced by:  isgrpi  17800  issubg2  17961  symggrp  18171  isdrngd  19129  psrgrp  19760  cnlmod  23310  dchrabl  25393  motgrp  25856  ldualgrplem  35221  tgrpgrplem  36825  erngdvlem1  37064  erngdvlem1-rN  37072  dvhgrp  37183  mendring  38606
  Copyright terms: Public domain W3C validator