Theorem isoas

Description: Congruence theorem for isocele triangles: if two angles of a triangle are congruent, then the corresponding sides also are. (Contributed by Thierry Arnoux, 5-Oct-2020)

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

Proof

Step	Hyp	Ref	Expression
1		isoas.p	$$⊢P={\mathrm{Base}}_G$$
2		isoas.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		isoas.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		isoas.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
5		isoas.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
6		isoas.a	$$⊢\phi \to A\in P$$
7		isoas.b	$$⊢\phi \to B\in P$$
8		isoas.c	$$⊢\phi \to C\in P$$
9		isoas.1	$$⊢\phi \to \neg \left(C\in \left(ALB\right)\vee A=B\right)$$
10		isoas.2	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“ACB”⟩$$
11		eqid	$$⊢{\sim }_{𝒢}\left(G\right)={\sim }_{𝒢}\left(G\right)$$
12	1 4 3 5 6 7 8 9	ncolrot1	$$⊢\phi \to \neg \left(A\in \left(BLC\right)\vee B=C\right)$$
13	1 2 3 5 7 8	axtgcgrrflx	$$⊢\phi \to B\underset{\dot{}}{-}C=C\underset{\dot{}}{-}B$$
14		eqid	$$⊢{hl}_{𝒢}\left(G\right)={hl}_{𝒢}\left(G\right)$$
15	1 3 5 14 6 7 8 6 8 7 10	cgracom	$$⊢\phi \to ⟨“ACB”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“ABC”⟩$$
16	1 3 2 5 6 8 7 6 7 8 15	cgraswaplr	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“CBA”⟩$$
17	1 2 3 5 7 8 6 8 7 6 4 12 13 16 10	tgasa	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}\left(G\right)⟨“CBA”⟩$$
18	1 2 3 11 5 7 8 6 8 7 6 17	cgr3simp3	$$⊢\phi \to A\underset{\dot{}}{-}B=A\underset{\dot{}}{-}C$$