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Theorem isgrpd2 18841
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 2732, but we make an exception for theorems such as isgrpd2 18841, ismndd 18646, and islmodd 20476 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 7415 . . . . 5 (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥))
87eqeq1d 2734 . . . 4 (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 ))
98rspcev 3612 . . 3 ((𝑁𝐵 ∧ (𝑁 + 𝑥) = 0 ) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
105, 6, 9syl2anc 584 . 2 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
111, 2, 3, 4, 10isgrpd2e 18840 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3070  cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  0gc0g 17384  Mndcmnd 18624  Grpcgrp 18818
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 2703
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 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-grp 18821
This theorem is referenced by:  prdsgrpd  18932  oppggrp  19223
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