Theorem ragtrivb

Description: Trivial right angle. Theorem 8.5 of Schwabhauser p. 58. (Contributed by Thierry Arnoux, 25-Aug-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	$$⊢\phi \to C\in P$$
Assertion	ragtrivb	$$⊢\phi \to ⟨“ABB”⟩\in {∟}_{𝒢}\left(G\right)$$

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		eqid	$$⊢S\left(B\right)=S\left(B\right)$$
11	1 2 3 4 5 6 8 10	mircinv	$$⊢\phi \to S\left(B\right)\left(B\right)=B$$
12	11	oveq2d	$$⊢\phi \to A\underset{\dot{}}{-}S\left(B\right)\left(B\right)=A\underset{\dot{}}{-}B$$
13	12	eqcomd	$$⊢\phi \to A\underset{\dot{}}{-}B=A\underset{\dot{}}{-}S\left(B\right)\left(B\right)$$
14	1 2 3 4 5 6 7 8 8	israg	$$⊢\phi \to \left(⟨“ABB”⟩\in {∟}_{𝒢}\left(G\right)\leftrightarrow A\underset{\dot{}}{-}B=A\underset{\dot{}}{-}S\left(B\right)\left(B\right)\right)$$
15	13 14	mpbird	$$⊢\phi \to ⟨“ABB”⟩\in {∟}_{𝒢}\left(G\right)$$