Theorem orngogrp 21113

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 2761	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2761	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2761	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2761	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 21111	. 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 1164	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3077   class class class wbr 5103  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  ._rcmulr 17422  lecple 17428  0_gc0g 17603  oGrpcogrp 20327  Ringcrg 20452  oRingcorng 21107

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6493  df-fv 6545  df-ov 7421  df-orng 21109

This theorem is used by:  orngsqr  21116  ornglmulle  21117  orngrmulle  21118  ofldtos  21123  suborng  21126  ofldchr  21875  isarchiofld  33753  nn0omnd  33898