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Theorem isgrpd2e 17922
Description: Deduce a group from its properties. In this version of isgrpd2 17923, 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 3125 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 )
4 isgrpd2.b . . . 4 (𝜑𝐵 = (Base‘𝐺))
5 isgrpd2.p . . . . . . 7 (𝜑+ = (+g𝐺))
65oveqd 6991 . . . . . 6 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐺)𝑥))
7 isgrpd2.z . . . . . 6 (𝜑0 = (0g𝐺))
86, 7eqeq12d 2786 . . . . 5 (𝜑 → ((𝑦 + 𝑥) = 0 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
94, 8rexeqbidv 3335 . . . 4 (𝜑 → (∃𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
104, 9raleqbidv 3334 . . 3 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
113, 10mpbid 224 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺))
12 eqid 2771 . . 3 (Base‘𝐺) = (Base‘𝐺)
13 eqid 2771 . . 3 (+g𝐺) = (+g𝐺)
14 eqid 2771 . . 3 (0g𝐺) = (0g𝐺)
1512, 13, 14isgrp 17909 . 2 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
161, 11, 15sylanbrc 575 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wcel 2051  wral 3081  wrex 3082  cfv 6185  (class class class)co 6974  Basecbs 16337  +gcplusg 16419  0gc0g 16567  Mndcmnd 17774  Grpcgrp 17903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-iota 6149  df-fv 6193  df-ov 6977  df-grp 17906
This theorem is referenced by:  isgrpd2  17923  isgrpde  17924
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