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Theorem isgrpd2e 17422
Description: Deduce a group from its properties. In this version of isgrpd2 17423, 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 2963 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 )
4 isgrpd2.b . . . 4 (𝜑𝐵 = (Base‘𝐺))
5 isgrpd2.p . . . . . . 7 (𝜑+ = (+g𝐺))
65oveqd 6652 . . . . . 6 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐺)𝑥))
7 isgrpd2.z . . . . . 6 (𝜑0 = (0g𝐺))
86, 7eqeq12d 2635 . . . . 5 (𝜑 → ((𝑦 + 𝑥) = 0 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
94, 8rexeqbidv 3148 . . . 4 (𝜑 → (∃𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
104, 9raleqbidv 3147 . . 3 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 0 ↔ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
113, 10mpbid 222 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺))
12 eqid 2620 . . 3 (Base‘𝐺) = (Base‘𝐺)
13 eqid 2620 . . 3 (+g𝐺) = (+g𝐺)
14 eqid 2620 . . 3 (0g𝐺) = (0g𝐺)
1512, 13, 14isgrp 17409 . 2 (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g𝐺)𝑥) = (0g𝐺)))
161, 11, 15sylanbrc 697 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  wrex 2910  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  0gc0g 16081  Mndcmnd 17275  Grpcgrp 17403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-grp 17406
This theorem is referenced by:  isgrpd2  17423  isgrpde  17424
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