Theorem iseqlgd

Description: Condition for a triangle to be equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020)

	Ref	Expression
Hypotheses	iseqlg.p	$$⊢P={\mathrm{Base}}_G$$
	iseqlg.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	iseqlg.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	iseqlg.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
	iseqlg.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	iseqlg.a	$$⊢\phi \to A\in P$$
	iseqlg.b	$$⊢\phi \to B\in P$$
	iseqlg.c	$$⊢\phi \to C\in P$$
	iseqlgd.1	$$⊢\phi \to A\underset{\dot{}}{-}B=B\underset{\dot{}}{-}C$$
	iseqlgd.2	$$⊢\phi \to B\underset{\dot{}}{-}C=C\underset{\dot{}}{-}A$$
	iseqlgd.3	$$⊢\phi \to C\underset{\dot{}}{-}A=A\underset{\dot{}}{-}B$$
Assertion	iseqlgd	$$⊢\phi \to ⟨“ABC”⟩\in {Equilaterals}_{𝒢}\left(G\right)$$

Proof

Step	Hyp	Ref	Expression
1		iseqlg.p	$$⊢P={\mathrm{Base}}_G$$
2		iseqlg.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		iseqlg.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		iseqlg.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
5		iseqlg.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
6		iseqlg.a	$$⊢\phi \to A\in P$$
7		iseqlg.b	$$⊢\phi \to B\in P$$
8		iseqlg.c	$$⊢\phi \to C\in P$$
9		iseqlgd.1	$$⊢\phi \to A\underset{\dot{}}{-}B=B\underset{\dot{}}{-}C$$
10		iseqlgd.2	$$⊢\phi \to B\underset{\dot{}}{-}C=C\underset{\dot{}}{-}A$$
11		iseqlgd.3	$$⊢\phi \to C\underset{\dot{}}{-}A=A\underset{\dot{}}{-}B$$
12		eqid	$$⊢{\sim }_{𝒢}\left(G\right)={\sim }_{𝒢}\left(G\right)$$
13	1 2 12 5 6 7 8 7 8 6 9 10 11	trgcgr	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}\left(G\right)⟨“BCA”⟩$$
14	1 2 3 4 5 6 7 8	iseqlg	$$⊢\phi \to \left(⟨“ABC”⟩\in {Equilaterals}_{𝒢}\left(G\right)\leftrightarrow ⟨“ABC”⟩{\sim }_{𝒢}\left(G\right)⟨“BCA”⟩\right)$$
15	13 14	mpbird	$$⊢\phi \to ⟨“ABC”⟩\in {Equilaterals}_{𝒢}\left(G\right)$$