Theorem lmimot

Description: Line mirroring is a motion of the geometric space. Theorem 10.11 of Schwabhauser p. 90. (Contributed by Thierry Arnoux, 15-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	lmimot	⊢ ( 𝜑 → 𝑀 ∈ ( 𝐺 Ismt 𝐺 ) )

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	lmif1o	⊢ ( 𝜑 → 𝑀 : 𝑃 –1-1-onto→ 𝑃 )
10	4	adantr	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝐺 ∈ TarskiG )
11	5	adantr	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝐺 DimTarskiG≥ 2 )
12	8	adantr	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝐷 ∈ ran 𝐿 )
13		simprl	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝑎 ∈ 𝑃 )
14		simprr	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝑏 ∈ 𝑃 )
15	1 2 3 10 11 6 7 12 13 14	lmiiso	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → ( ( 𝑀 ‘ 𝑎 ) − ( 𝑀 ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) )
16	15	ralrimivva	⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑃 ∀ 𝑏 ∈ 𝑃 ( ( 𝑀 ‘ 𝑎 ) − ( 𝑀 ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) )
17	1 2	ismot	⊢ ( 𝐺 ∈ TarskiG → ( 𝑀 ∈ ( 𝐺 Ismt 𝐺 ) ↔ ( 𝑀 : 𝑃 –1-1-onto→ 𝑃 ∧ ∀ 𝑎 ∈ 𝑃 ∀ 𝑏 ∈ 𝑃 ( ( 𝑀 ‘ 𝑎 ) − ( 𝑀 ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) ) ) )
18	4 17	syl	⊢ ( 𝜑 → ( 𝑀 ∈ ( 𝐺 Ismt 𝐺 ) ↔ ( 𝑀 : 𝑃 –1-1-onto→ 𝑃 ∧ ∀ 𝑎 ∈ 𝑃 ∀ 𝑏 ∈ 𝑃 ( ( 𝑀 ‘ 𝑎 ) − ( 𝑀 ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) ) ) )
19	9 16 18	mpbir2and	⊢ ( 𝜑 → 𝑀 ∈ ( 𝐺 Ismt 𝐺 ) )