Theorem cgraswaplr

Description: Swap both side of angle congruence. (Contributed by Thierry Arnoux, 5-Oct-2020)

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

Proof

Step	Hyp	Ref	Expression
1		cgracol.p	$$⊢P={\mathrm{Base}}_G$$
2		cgracol.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
3		cgracol.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
4		cgracol.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		cgracol.a	$$⊢\phi \to A\in P$$
6		cgracol.b	$$⊢\phi \to B\in P$$
7		cgracol.c	$$⊢\phi \to C\in P$$
8		cgracol.d	$$⊢\phi \to D\in P$$
9		cgracol.e	$$⊢\phi \to E\in P$$
10		cgracol.f	$$⊢\phi \to F\in P$$
11		cgracol.1	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEF”⟩$$
12		eqid	$$⊢{hl}_{𝒢}\left(G\right)={hl}_{𝒢}\left(G\right)$$
13	1 2 12 4 5 6 7 8 9 10 11	cgrane2	$$⊢\phi \to B\ne C$$
14	13	necomd	$$⊢\phi \to C\ne B$$
15	1 2 12 4 5 6 7 8 9 10 11	cgrane1	$$⊢\phi \to A\ne B$$
16	15	necomd	$$⊢\phi \to B\ne A$$
17	1 2 4 12 7 6 5 14 16	cgraswap	$$⊢\phi \to ⟨“CBA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“ABC”⟩$$
18	1 2 4 12 7 6 5 5 6 7 17 8 9 10 11	cgratr	$$⊢\phi \to ⟨“CBA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEF”⟩$$
19	1 2 12 4 5 6 7 8 9 10 11	cgrane3	$$⊢\phi \to E\ne D$$
20	19	necomd	$$⊢\phi \to D\ne E$$
21	1 2 12 4 5 6 7 8 9 10 11	cgrane4	$$⊢\phi \to E\ne F$$
22	1 2 4 12 8 9 10 20 21	cgraswap	$$⊢\phi \to ⟨“DEF”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“FED”⟩$$
23	1 2 4 12 7 6 5 8 9 10 18 10 9 8 22	cgratr	$$⊢\phi \to ⟨“CBA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“FED”⟩$$