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Theorem isgrpd2e 17264
 Description: Deduce a group from its properties. In this version of isgrpd2 17265, 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 2949 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 )
4 isgrpd2.b . . . 4 (𝜑𝐵 = (Base‘𝐺))
5 isgrpd2.p . . . . . . 7 (𝜑+ = (+g𝐺))
65oveqd 6566 . . . . . 6 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐺)𝑥))
7 isgrpd2.z . . . . . 6 (𝜑0 = (0g𝐺))
86, 7eqeq12d 2625 . . . . 5 (𝜑 → ((𝑦 + 𝑥) = 0 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
94, 8rexeqbidv 3130 . . . 4 (𝜑 → (∃𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
104, 9raleqbidv 3129 . . 3 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
113, 10mpbid 221 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺))
12 eqid 2610 . . 3 (Base‘𝐺) = (Base‘𝐺)
13 eqid 2610 . . 3 (+g𝐺) = (+g𝐺)
14 eqid 2610 . . 3 (0g𝐺) = (0g𝐺)
1512, 13, 14isgrp 17251 . 2 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
161, 11, 15sylanbrc 695 1 (𝜑𝐺 ∈ Grp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Mndcmnd 17117  Grpcgrp 17245 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-grp 17248 This theorem is referenced by:  isgrpd2  17265  isgrpde  17266
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