Theorem orng0le1

Description: In an ordered ring, the ring unit is positive. (Contributed by Thierry Arnoux, 21-Jan-2018)

	Ref	Expression
Hypotheses	orng0le1.1	$$⊢\underset{\dot{}}{0}=0_F$$
	orng0le1.2	$$⊢\underset{\dot{}}{1}=1_F$$
	orng0le1.3	$$⊢\underset{\dot{}}{\le }={\le }_F$$
Assertion	orng0le1	$$⊢F\in \mathrm{oRing}\to \underset{\dot{}}{0}\underset{\dot{}}{\le }\underset{\dot{}}{1}$$

Proof

Step	Hyp	Ref	Expression
1		orng0le1.1	$$⊢\underset{\dot{}}{0}=0_F$$
2		orng0le1.2	$$⊢\underset{\dot{}}{1}=1_F$$
3		orng0le1.3	$$⊢\underset{\dot{}}{\le }={\le }_F$$
4		orngring	$$⊢F\in \mathrm{oRing}\to F\in \mathrm{Ring}$$
5		eqid	$$⊢{\mathrm{Base}}_F={\mathrm{Base}}_F$$
6	5 2	ringidcl	$$⊢F\in \mathrm{Ring}\to \underset{\dot{}}{1}\in {\mathrm{Base}}_F$$
7	4 6	syl	$$⊢F\in \mathrm{oRing}\to \underset{\dot{}}{1}\in {\mathrm{Base}}_F$$
8		eqid	$$⊢{\cdot }_F={\cdot }_F$$
9	5 3 1 8	orngsqr	$$⊢\left(F\in \mathrm{oRing}\wedge \underset{\dot{}}{1}\in {\mathrm{Base}}_F\right)\to \underset{\dot{}}{0}\underset{\dot{}}{\le }\left(\underset{\dot{}}{1}{\cdot }_F\underset{\dot{}}{1}\right)$$
10	7 9	mpdan	$$⊢F\in \mathrm{oRing}\to \underset{\dot{}}{0}\underset{\dot{}}{\le }\left(\underset{\dot{}}{1}{\cdot }_F\underset{\dot{}}{1}\right)$$
11	5 8 2	ringlidm	$$⊢\left(F\in \mathrm{Ring}\wedge \underset{\dot{}}{1}\in {\mathrm{Base}}_F\right)\to \underset{\dot{}}{1}{\cdot }_F\underset{\dot{}}{1}=\underset{\dot{}}{1}$$
12	4 6 11	syl2anc2	$$⊢F\in \mathrm{oRing}\to \underset{\dot{}}{1}{\cdot }_F\underset{\dot{}}{1}=\underset{\dot{}}{1}$$
13	10 12	breqtrd	$$⊢F\in \mathrm{oRing}\to \underset{\dot{}}{0}\underset{\dot{}}{\le }\underset{\dot{}}{1}$$