Theorem trgcgr

Description: Triangle congruence. (Contributed by Thierry Arnoux, 27-Apr-2019)

	Ref	Expression
Hypotheses	trgcgrg.p	$$⊢P={\mathrm{Base}}_G$$
	trgcgrg.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	trgcgrg.r	$$⊢\underset{\dot{}}{\sim }={\sim }_{𝒢}\left(G\right)$$
	trgcgrg.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	trgcgrg.a	$$⊢\phi \to A\in P$$
	trgcgrg.b	$$⊢\phi \to B\in P$$
	trgcgrg.c	$$⊢\phi \to C\in P$$
	trgcgrg.d	$$⊢\phi \to D\in P$$
	trgcgrg.e	$$⊢\phi \to E\in P$$
	trgcgrg.f	$$⊢\phi \to F\in P$$
	trgcgr.1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
	trgcgr.2	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$
	trgcgr.3	$$⊢\phi \to C\underset{\dot{}}{-}A=F\underset{\dot{}}{-}D$$
Assertion	trgcgr	$$⊢\phi \to ⟨“ABC”⟩\underset{\dot{}}{\sim }⟨“DEF”⟩$$

Proof

Step	Hyp	Ref	Expression
1		trgcgrg.p	$$⊢P={\mathrm{Base}}_G$$
2		trgcgrg.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		trgcgrg.r	$$⊢\underset{\dot{}}{\sim }={\sim }_{𝒢}\left(G\right)$$
4		trgcgrg.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		trgcgrg.a	$$⊢\phi \to A\in P$$
6		trgcgrg.b	$$⊢\phi \to B\in P$$
7		trgcgrg.c	$$⊢\phi \to C\in P$$
8		trgcgrg.d	$$⊢\phi \to D\in P$$
9		trgcgrg.e	$$⊢\phi \to E\in P$$
10		trgcgrg.f	$$⊢\phi \to F\in P$$
11		trgcgr.1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
12		trgcgr.2	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$
13		trgcgr.3	$$⊢\phi \to C\underset{\dot{}}{-}A=F\underset{\dot{}}{-}D$$
14	1 2 3 4 5 6 7 8 9 10	trgcgrg	$$⊢\phi \to \left(⟨“ABC”⟩\underset{\dot{}}{\sim }⟨“DEF”⟩\leftrightarrow \left(A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E\wedge B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F\wedge C\underset{\dot{}}{-}A=F\underset{\dot{}}{-}D\right)\right)$$
15	11 12 13 14	mpbir3and	$$⊢\phi \to ⟨“ABC”⟩\underset{\dot{}}{\sim }⟨“DEF”⟩$$