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Theorem isgrpd2e 18114
Description: Deduce a group from its properties. In this version of isgrpd2 18115, we don't assume there is an expression for the inverse of 𝑥. (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)
isgrpd2e.n ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
Assertion
Ref Expression
isgrpd2e (𝜑𝐺 ∈ Grp)
Distinct variable groups:   𝑥,𝑦, +   𝑦, 0   𝑥,𝐵,𝑦   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦
Allowed substitution hint:   0 (𝑥)

Proof of Theorem isgrpd2e
StepHypRef Expression
1 isgrpd2.g . 2 (𝜑𝐺 ∈ Mnd)
2 isgrpd2e.n . . . 4 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
32ralrimiva 3149 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 )
4 isgrpd2.b . . . 4 (𝜑𝐵 = (Base‘𝐺))
5 isgrpd2.p . . . . . . 7 (𝜑+ = (+g𝐺))
65oveqd 7152 . . . . . 6 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐺)𝑥))
7 isgrpd2.z . . . . . 6 (𝜑0 = (0g𝐺))
86, 7eqeq12d 2814 . . . . 5 (𝜑 → ((𝑦 + 𝑥) = 0 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
94, 8rexeqbidv 3355 . . . 4 (𝜑 → (∃𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
104, 9raleqbidv 3354 . . 3 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
113, 10mpbid 235 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺))
12 eqid 2798 . . 3 (Base‘𝐺) = (Base‘𝐺)
13 eqid 2798 . . 3 (+g𝐺) = (+g𝐺)
14 eqid 2798 . . 3 (0g𝐺) = (0g𝐺)
1512, 13, 14isgrp 18101 . 2 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
161, 11, 15sylanbrc 586 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  wral 3106  wrex 3107  cfv 6324  (class class class)co 7135  Basecbs 16475  +gcplusg 16557  0gc0g 16705  Mndcmnd 17903  Grpcgrp 18095
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-grp 18098
This theorem is referenced by:  isgrpd2  18115  isgrpde  18116
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