Theorem lmieq

Description: Equality deduction for line mirroring. Theorem 10.7 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	$$⊢\phi \to D\in \mathrm{ran}L$$
	lmicl.1	$$⊢\phi \to A\in P$$
	lmieq.c	$$⊢\phi \to B\in P$$
	lmieq.d	$$⊢\phi \to M\left(A\right)=M\left(B\right)$$
Assertion	lmieq	$$⊢\phi \to A=B$$

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		lmicl.1	$$⊢\phi \to A\in P$$
10		lmieq.c	$$⊢\phi \to B\in P$$
11		lmieq.d	$$⊢\phi \to M\left(A\right)=M\left(B\right)$$
12		fveqeq2	$$⊢b=A\to \left(M\left(b\right)=M\left(B\right)\leftrightarrow M\left(A\right)=M\left(B\right)\right)$$
13		fveqeq2	$$⊢b=B\to \left(M\left(b\right)=M\left(B\right)\leftrightarrow M\left(B\right)=M\left(B\right)\right)$$
14	1 2 3 4 5 6 7 8 10	lmicl	$$⊢\phi \to M\left(B\right)\in P$$
15	1 2 3 4 5 6 7 8 14	lmireu	$$⊢\phi \to \exists !b\in PM\left(b\right)=M\left(B\right)$$
16		eqidd	$$⊢\phi \to M\left(B\right)=M\left(B\right)$$
17	12 13 15 9 10 11 16	reu2eqd	$$⊢\phi \to A=B$$