Theorem lmif1o

Description: The line mirroring function M is a bijection. Theorem 10.9 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	$$⊢\phi \to D\in \mathrm{ran}L$$
Assertion	lmif1o	$$⊢\phi \to M:P\underset{1-1\; onto}{⟶}P$$

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	1 2 3 4 5 6 7 8	lmif	$$⊢\phi \to M:P⟶P$$
10	9	ffnd	$$⊢\phi \to MFnP$$
11	4	adantr	$$⊢\left(\phi \wedge b\in P\right)\to G\in {𝒢}_{\mathrm{Tarski}}$$
12	5	adantr	$$⊢\left(\phi \wedge b\in P\right)\to G{\mathrm{Dim}}_{𝒢}\ge 2$$
13	8	adantr	$$⊢\left(\phi \wedge b\in P\right)\to D\in \mathrm{ran}L$$
14		simpr	$$⊢\left(\phi \wedge b\in P\right)\to b\in P$$
15	1 2 3 11 12 6 7 13 14	lmilmi	$$⊢\left(\phi \wedge b\in P\right)\to M\left(M\left(b\right)\right)=b$$
16	15	ralrimiva	$$⊢\phi \to \forall b\in PM\left(M\left(b\right)\right)=b$$
17		nvocnv	$$⊢\left(M:P⟶P\wedge \forall b\in PM\left(M\left(b\right)\right)=b\right)\to M^{-1}=M$$
18	9 16 17	syl2anc	$$⊢\phi \to M^{-1}=M$$
19		nvof1o	$$⊢\left(MFnP\wedge M^{-1}=M\right)\to M:P\underset{1-1\; onto}{⟶}P$$
20	10 18 19	syl2anc	$$⊢\phi \to M:P\underset{1-1\; onto}{⟶}P$$