Theorem lmicinv

Description: The mirroring line is an invariant. (Contributed by Thierry Arnoux, 8-Aug-2020)

	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	$$⊢\phi \to D\in \mathrm{ran}L$$
	lmicl.1	$$⊢\phi \to A\in P$$
	lmicinv.1	$$⊢\phi \to A\in D$$
Assertion	lmicinv	$$⊢\phi \to M\left(A\right)=A$$

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		lmicl.1	$$⊢\phi \to A\in P$$
10		lmicinv.1	$$⊢\phi \to A\in D$$
11	1 2 3 4 5 6 7 8 9	lmiinv	$$⊢\phi \to \left(M\left(A\right)=A\leftrightarrow A\in D\right)$$
12	10 11	mpbird	$$⊢\phi \to M\left(A\right)=A$$