Theorem ragcom

Description: Commutative rule for right angles. Theorem 8.2 of Schwabhauser p. 57. (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	$$⊢\phi \to C\in P$$
	ragcom.1	$$⊢\phi \to ⟨“ABC”⟩\in {∟}_{𝒢}\left(G\right)$$
Assertion	ragcom	$$⊢\phi \to ⟨“CBA”⟩\in {∟}_{𝒢}\left(G\right)$$

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		ragcom.1	$$⊢\phi \to ⟨“ABC”⟩\in {∟}_{𝒢}\left(G\right)$$
11		eqid	$$⊢S\left(B\right)=S\left(B\right)$$
12	1 2 3 4 5 6 8 11 9	mircl	$$⊢\phi \to S\left(B\right)\left(C\right)\in P$$
13	1 2 3 4 5 6 7 8 9	israg	$$⊢\phi \to \left(⟨“ABC”⟩\in {∟}_{𝒢}\left(G\right)\leftrightarrow A\underset{\dot{}}{-}C=A\underset{\dot{}}{-}S\left(B\right)\left(C\right)\right)$$
14	10 13	mpbid	$$⊢\phi \to A\underset{\dot{}}{-}C=A\underset{\dot{}}{-}S\left(B\right)\left(C\right)$$
15	1 2 3 6 7 9 7 12 14	tgcgrcomlr	$$⊢\phi \to C\underset{\dot{}}{-}A=S\left(B\right)\left(C\right)\underset{\dot{}}{-}A$$
16	1 2 3 4 5 6 8 11 12 7	miriso	$$⊢\phi \to S\left(B\right)\left(S\left(B\right)\left(C\right)\right)\underset{\dot{}}{-}S\left(B\right)\left(A\right)=S\left(B\right)\left(C\right)\underset{\dot{}}{-}A$$
17	1 2 3 4 5 6 8 11 9	mirmir	$$⊢\phi \to S\left(B\right)\left(S\left(B\right)\left(C\right)\right)=C$$
18	17	oveq1d	$$⊢\phi \to S\left(B\right)\left(S\left(B\right)\left(C\right)\right)\underset{\dot{}}{-}S\left(B\right)\left(A\right)=C\underset{\dot{}}{-}S\left(B\right)\left(A\right)$$
19	15 16 18	3eqtr2d	$$⊢\phi \to C\underset{\dot{}}{-}A=C\underset{\dot{}}{-}S\left(B\right)\left(A\right)$$
20	1 2 3 4 5 6 9 8 7	israg	$$⊢\phi \to \left(⟨“CBA”⟩\in {∟}_{𝒢}\left(G\right)\leftrightarrow C\underset{\dot{}}{-}A=C\underset{\dot{}}{-}S\left(B\right)\left(A\right)\right)$$
21	19 20	mpbird	$$⊢\phi \to ⟨“CBA”⟩\in {∟}_{𝒢}\left(G\right)$$