Theorem orngogrp 30925

Description: An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.)

Assertion

Ref	Expression
orngogrp	⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)

Proof of Theorem orngogrp

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2798	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2798	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2798	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2798	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 30923	. 2 ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ (Base‘𝑅)∀𝑏 ∈ (Base‘𝑅)(((0g‘𝑅)(le‘𝑅)𝑎 ∧ (0g‘𝑅)(le‘𝑅)𝑏) → (0g‘𝑅)(le‘𝑅)(𝑎(.r‘𝑅)𝑏))))
6	5	simp2bi 1143	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  ∀wral 3106   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  ._rcmulr 16558  lecple 16564  0_gc0g 16705  Ringcrg 19290  oGrpcogrp 30749  oRingcorng 30919

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  ax-nul 5174

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-mo 2598  df-eu 2629  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-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  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-orng 30921

This theorem is referenced by:  orngsqr  30928  ornglmulle  30929  orngrmulle  30930  ofldtos  30935  ofldchr  30938  suborng  30939  isarchiofld  30941  nn0omnd  30965