Theorem tgsss1

Description: Third congruence theorem: SSS (Side-Side-Side): If the three pairs of sides of two triangles are equal in length, then the triangles are congruent. Theorem 11.51 of Schwabhauser p. 109. (Contributed by Thierry Arnoux, 1-Aug-2020)

	Ref	Expression
Hypotheses	tgsas.p	$$⊢P={\mathrm{Base}}_G$$
	tgsas.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	tgsas.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tgsas.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgsas.a	$$⊢\phi \to A\in P$$
	tgsas.b	$$⊢\phi \to B\in P$$
	tgsas.c	$$⊢\phi \to C\in P$$
	tgsas.d	$$⊢\phi \to D\in P$$
	tgsas.e	$$⊢\phi \to E\in P$$
	tgsas.f	$$⊢\phi \to F\in P$$
	tgsss.1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
	tgsss.2	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$
	tgsss.3	$$⊢\phi \to C\underset{\dot{}}{-}A=F\underset{\dot{}}{-}D$$
	tgsss.4	$$⊢\phi \to A\ne B$$
	tgsss.5	$$⊢\phi \to B\ne C$$
	tgsss.6	$$⊢\phi \to C\ne A$$
Assertion	tgsss1	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEF”⟩$$

Proof

Step	Hyp	Ref	Expression
1		tgsas.p	$$⊢P={\mathrm{Base}}_G$$
2		tgsas.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		tgsas.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tgsas.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		tgsas.a	$$⊢\phi \to A\in P$$
6		tgsas.b	$$⊢\phi \to B\in P$$
7		tgsas.c	$$⊢\phi \to C\in P$$
8		tgsas.d	$$⊢\phi \to D\in P$$
9		tgsas.e	$$⊢\phi \to E\in P$$
10		tgsas.f	$$⊢\phi \to F\in P$$
11		tgsss.1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
12		tgsss.2	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$
13		tgsss.3	$$⊢\phi \to C\underset{\dot{}}{-}A=F\underset{\dot{}}{-}D$$
14		tgsss.4	$$⊢\phi \to A\ne B$$
15		tgsss.5	$$⊢\phi \to B\ne C$$
16		tgsss.6	$$⊢\phi \to C\ne A$$
17		eqid	$$⊢{hl}_{𝒢}\left(G\right)={hl}_{𝒢}\left(G\right)$$
18		eqid	$$⊢{\sim }_{𝒢}\left(G\right)={\sim }_{𝒢}\left(G\right)$$
19	1 2 18 4 5 6 7 8 9 10 11 12 13	trgcgr	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}\left(G\right)⟨“DEF”⟩$$
20	1 3 4 17 5 6 7 8 9 10 14 15 19	cgrcgra	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEF”⟩$$