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Theorem isgrpd 18777
Description: Deduce a group from its properties. Unlike isgrpd2 18775, 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 7365 . . . . 5 (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥))
109eqeq1d 2735 . . . 4 (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 ))
1110rspcev 3580 . . 3 ((𝑁𝐵 ∧ (𝑁 + 𝑥) = 0 ) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
127, 8, 11syl2anc 585 . 2 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
131, 2, 3, 4, 5, 6, 12isgrpde 18776 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3070  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Grpcgrp 18753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-riota 7314  df-ov 7361  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756
This theorem is referenced by:  isgrpi  18778  issubg2  18948  symggrp  19187  isdrngd  20226  isdrngdOLD  20228  psrgrpOLD  21383  cnlmod  24519  dchrabl  26618  motgrp  27527  ldualgrplem  37653  tgrpgrplem  39258  erngdvlem1  39497  erngdvlem1-rN  39505  dvhgrp  39616  mendring  41562
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