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Theorem isgrpd2e 18926
Description: Deduce a group from its properties. In this version of isgrpd2 18927, 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 3143 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 )
4 isgrpd2.b . . . 4 (𝜑𝐵 = (Base‘𝐺))
5 isgrpd2.p . . . . . . 7 (𝜑+ = (+g𝐺))
65oveqd 7443 . . . . . 6 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐺)𝑥))
7 isgrpd2.z . . . . . 6 (𝜑0 = (0g𝐺))
86, 7eqeq12d 2744 . . . . 5 (𝜑 → ((𝑦 + 𝑥) = 0 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
94, 8rexeqbidv 3341 . . . 4 (𝜑 → (∃𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
104, 9raleqbidv 3340 . . 3 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
113, 10mpbid 231 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺))
12 eqid 2728 . . 3 (Base‘𝐺) = (Base‘𝐺)
13 eqid 2728 . . 3 (+g𝐺) = (+g𝐺)
14 eqid 2728 . . 3 (0g𝐺) = (0g𝐺)
1512, 13, 14isgrp 18910 . 2 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
161, 11, 15sylanbrc 581 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3058  wrex 3067  cfv 6553  (class class class)co 7426  Basecbs 17189  +gcplusg 17242  0gc0g 17430  Mndcmnd 18703  Grpcgrp 18904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-grp 18907
This theorem is referenced by:  isgrpd2  18927  isgrpde  18928  rloccring  33017
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