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Theorem isgrpd2 18775
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 2733, but we make an exception for theorems such as isgrpd2 18775, ismndd 18583, and islmodd 20342 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 7365 . . . . 5 (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥))
87eqeq1d 2735 . . . 4 (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 ))
98rspcev 3580 . . 3 ((𝑁𝐵 ∧ (𝑁 + 𝑥) = 0 ) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
105, 6, 9syl2anc 585 . 2 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
111, 2, 3, 4, 10isgrpd2e 18774 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wrex 3070  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  Mndcmnd 18561  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-ext 2704
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  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-iota 6449  df-fv 6505  df-ov 7361  df-grp 18756
This theorem is referenced by:  prdsgrpd  18862  oppggrp  19143
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