Theorem ragflat2

Description: Deduce equality from two right angles. Theorem 8.6 of Schwabhauser p. 58. (Contributed by Thierry Arnoux, 3-Sep-2019)

	Ref	Expression
Hypotheses	israg.p	$$⊢P={\mathrm{Base}}_G$$
	israg.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	israg.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	israg.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
	israg.s	$$⊢S={pInv}_{𝒢}\left(G\right)$$
	israg.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	israg.a	$$⊢\phi \to A\in P$$
	israg.b	$$⊢\phi \to B\in P$$
	israg.c	$$⊢\phi \to C\in P$$
	ragflat2.d	$$⊢\phi \to D\in P$$
	ragflat2.1	$$⊢\phi \to ⟨“ABC”⟩\in {∟}_{𝒢}\left(G\right)$$
	ragflat2.2	$$⊢\phi \to ⟨“DBC”⟩\in {∟}_{𝒢}\left(G\right)$$
	ragflat2.3	$$⊢\phi \to C\in \left(AID\right)$$
Assertion	ragflat2	$$⊢\phi \to B=C$$

Proof

Step	Hyp	Ref	Expression
1		israg.p	$$⊢P={\mathrm{Base}}_G$$
2		israg.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		israg.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		israg.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
5		israg.s	$$⊢S={pInv}_{𝒢}\left(G\right)$$
6		israg.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
7		israg.a	$$⊢\phi \to A\in P$$
8		israg.b	$$⊢\phi \to B\in P$$
9		israg.c	$$⊢\phi \to C\in P$$
10		ragflat2.d	$$⊢\phi \to D\in P$$
11		ragflat2.1	$$⊢\phi \to ⟨“ABC”⟩\in {∟}_{𝒢}\left(G\right)$$
12		ragflat2.2	$$⊢\phi \to ⟨“DBC”⟩\in {∟}_{𝒢}\left(G\right)$$
13		ragflat2.3	$$⊢\phi \to C\in \left(AID\right)$$
14		eqid	$$⊢{\sim }_{𝒢}\left(G\right)={\sim }_{𝒢}\left(G\right)$$
15		eqid	$$⊢S\left(B\right)=S\left(B\right)$$
16	1 2 3 4 5 6 8 15 9	mircl	$$⊢\phi \to S\left(B\right)\left(C\right)\in P$$
17	1 2 3 4 5 6 7 8 9	israg	$$⊢\phi \to \left(⟨“ABC”⟩\in {∟}_{𝒢}\left(G\right)\leftrightarrow A\underset{\dot{}}{-}C=A\underset{\dot{}}{-}S\left(B\right)\left(C\right)\right)$$
18	11 17	mpbid	$$⊢\phi \to A\underset{\dot{}}{-}C=A\underset{\dot{}}{-}S\left(B\right)\left(C\right)$$
19	1 2 3 4 5 6 10 8 9	israg	$$⊢\phi \to \left(⟨“DBC”⟩\in {∟}_{𝒢}\left(G\right)\leftrightarrow D\underset{\dot{}}{-}C=D\underset{\dot{}}{-}S\left(B\right)\left(C\right)\right)$$
20	12 19	mpbid	$$⊢\phi \to D\underset{\dot{}}{-}C=D\underset{\dot{}}{-}S\left(B\right)\left(C\right)$$
21	1 4 3 6 7 10 9 14 16 7 2 13 18 20	tgidinside	$$⊢\phi \to C=S\left(B\right)\left(C\right)$$
22	21	eqcomd	$$⊢\phi \to S\left(B\right)\left(C\right)=C$$
23	1 2 3 4 5 6 8 15 9	mirinv	$$⊢\phi \to \left(S\left(B\right)\left(C\right)=C\leftrightarrow B=C\right)$$
24	22 23	mpbid	$$⊢\phi \to B=C$$