Theorem lmiiso

Description: The line mirroring function is an isometry, i.e. it is conserves congruence. Because it is also a bijection, it is also a motion. Theorem 10.10 of Schwabhauser p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019)

	Ref	Expression
Hypotheses	ismid.p	$$⊢P={\mathrm{Base}}_G$$
	ismid.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	ismid.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	ismid.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	ismid.1	$$⊢\phi \to G{\mathrm{Dim}}_{𝒢}\ge 2$$
	lmif.m	$$⊢M={lInv}_{𝒢}\left(G\right)\left(D\right)$$
	lmif.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
	lmif.d	$$⊢\phi \to D\in \mathrm{ran}L$$
	lmiiso.1	$$⊢\phi \to A\in P$$
	lmiiso.2	$$⊢\phi \to B\in P$$
Assertion	lmiiso	$$⊢\phi \to M\left(A\right)\underset{\dot{}}{-}M\left(B\right)=A\underset{\dot{}}{-}B$$

Proof

Step	Hyp	Ref	Expression
1		ismid.p	$$⊢P={\mathrm{Base}}_G$$
2		ismid.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		ismid.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		ismid.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		ismid.1	$$⊢\phi \to G{\mathrm{Dim}}_{𝒢}\ge 2$$
6		lmif.m	$$⊢M={lInv}_{𝒢}\left(G\right)\left(D\right)$$
7		lmif.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
8		lmif.d	$$⊢\phi \to D\in \mathrm{ran}L$$
9		lmiiso.1	$$⊢\phi \to A\in P$$
10		lmiiso.2	$$⊢\phi \to B\in P$$
11		eqid	$$⊢{pInv}_{𝒢}\left(G\right)\left(\left(A{mid}_{𝒢}\left(G\right)M\left(A\right)\right){mid}_{𝒢}\left(G\right)\left(B{mid}_{𝒢}\left(G\right)M\left(B\right)\right)\right)={pInv}_{𝒢}\left(G\right)\left(\left(A{mid}_{𝒢}\left(G\right)M\left(A\right)\right){mid}_{𝒢}\left(G\right)\left(B{mid}_{𝒢}\left(G\right)M\left(B\right)\right)\right)$$
12		eqid	$$⊢\left(A{mid}_{𝒢}\left(G\right)M\left(A\right)\right){mid}_{𝒢}\left(G\right)\left(B{mid}_{𝒢}\left(G\right)M\left(B\right)\right)=\left(A{mid}_{𝒢}\left(G\right)M\left(A\right)\right){mid}_{𝒢}\left(G\right)\left(B{mid}_{𝒢}\left(G\right)M\left(B\right)\right)$$
13	1 2 3 4 5 6 7 8 9 10 11 12	lmiisolem	$$⊢\phi \to M\left(A\right)\underset{\dot{}}{-}M\left(B\right)=A\underset{\dot{}}{-}B$$