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 = ( Base ‘ 𝐺 )
	ismid.d	⊢ − = ( dist ‘ 𝐺 )
	ismid.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	ismid.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	ismid.1	⊢ ( 𝜑 → 𝐺 DimTarskiG≥ 2 )
	lmif.m	⊢ 𝑀 = ( ( lInvG ‘ 𝐺 ) ‘ 𝐷 )
	lmif.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	lmif.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
Assertion	lmif1o	⊢ ( 𝜑 → 𝑀 : 𝑃 –1-1-onto→ 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		ismid.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		ismid.d	⊢ − = ( dist ‘ 𝐺 )
3		ismid.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		ismid.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		ismid.1	⊢ ( 𝜑 → 𝐺 DimTarskiG≥ 2 )
6		lmif.m	⊢ 𝑀 = ( ( lInvG ‘ 𝐺 ) ‘ 𝐷 )
7		lmif.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
8		lmif.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
9	1 2 3 4 5 6 7 8	lmif	⊢ ( 𝜑 → 𝑀 : 𝑃 ⟶ 𝑃 )
10	9	ffnd	⊢ ( 𝜑 → 𝑀 Fn 𝑃 )
11	4	adantr	⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) → 𝐺 ∈ TarskiG )
12	5	adantr	⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) → 𝐺 DimTarskiG≥ 2 )
13	8	adantr	⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) → 𝐷 ∈ ran 𝐿 )
14		simpr	⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) → 𝑏 ∈ 𝑃 )
15	1 2 3 11 12 6 7 13 14	lmilmi	⊢ ( ( 𝜑 ∧ 𝑏 ∈ 𝑃 ) → ( 𝑀 ‘ ( 𝑀 ‘ 𝑏 ) ) = 𝑏 )
16	15	ralrimiva	⊢ ( 𝜑 → ∀ 𝑏 ∈ 𝑃 ( 𝑀 ‘ ( 𝑀 ‘ 𝑏 ) ) = 𝑏 )
17		nvocnv	⊢ ( ( 𝑀 : 𝑃 ⟶ 𝑃 ∧ ∀ 𝑏 ∈ 𝑃 ( 𝑀 ‘ ( 𝑀 ‘ 𝑏 ) ) = 𝑏 ) → ◡ 𝑀 = 𝑀 )
18	9 16 17	syl2anc	⊢ ( 𝜑 → ◡ 𝑀 = 𝑀 )
19		nvof1o	⊢ ( ( 𝑀 Fn 𝑃 ∧ ◡ 𝑀 = 𝑀 ) → 𝑀 : 𝑃 –1-1-onto→ 𝑃 )
20	10 18 19	syl2anc	⊢ ( 𝜑 → 𝑀 : 𝑃 –1-1-onto→ 𝑃 )