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Theorem isgrpd2 18817
Description: Deduce a group from its properties. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). Note: normally we don't use a 𝜑 antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2731, but we make an exception for theorems such as isgrpd2 18817, ismndd 18624, and islmodd 20426 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)
Hypotheses
Ref Expression
isgrpd2.b (𝜑𝐵 = (Base‘𝐺))
isgrpd2.p (𝜑+ = (+g𝐺))
isgrpd2.z (𝜑0 = (0g𝐺))
isgrpd2.g (𝜑𝐺 ∈ Mnd)
isgrpd2.n ((𝜑𝑥𝐵) → 𝑁𝐵)
isgrpd2.j ((𝜑𝑥𝐵) → (𝑁 + 𝑥) = 0 )
Assertion
Ref Expression
isgrpd2 (𝜑𝐺 ∈ Grp)
Distinct variable groups:   𝑥, +   𝑥,𝐵   𝑥,𝐺   𝜑,𝑥
Allowed substitution hints:   𝑁(𝑥)   0 (𝑥)

Proof of Theorem isgrpd2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 isgrpd2.b . 2 (𝜑𝐵 = (Base‘𝐺))
2 isgrpd2.p . 2 (𝜑+ = (+g𝐺))
3 isgrpd2.z . 2 (𝜑0 = (0g𝐺))
4 isgrpd2.g . 2 (𝜑𝐺 ∈ Mnd)
5 isgrpd2.n . . 3 ((𝜑𝑥𝐵) → 𝑁𝐵)
6 isgrpd2.j . . 3 ((𝜑𝑥𝐵) → (𝑁 + 𝑥) = 0 )
7 oveq1 7400 . . . . 5 (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥))
87eqeq1d 2733 . . . 4 (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 ))
98rspcev 3609 . . 3 ((𝑁𝐵 ∧ (𝑁 + 𝑥) = 0 ) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
105, 6, 9syl2anc 584 . 2 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
111, 2, 3, 4, 10isgrpd2e 18816 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3069  cfv 6532  (class class class)co 7393  Basecbs 17126  +gcplusg 17179  0gc0g 17367  Mndcmnd 18602  Grpcgrp 18794
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-iota 6484  df-fv 6540  df-ov 7396  df-grp 18797
This theorem is referenced by:  prdsgrpd  18907  oppggrp  19188
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