Theorem lmiinv

Description: The invariants of the line mirroring lie on the mirror line. Theorem 10.8 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 )
	lmicl.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
Assertion	lmiinv	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝐴 ) = 𝐴 ↔ 𝐴 ∈ 𝐷 ) )

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		lmicl.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
10	1 2 3 4 5 6 7 8 9 9	islmib	⊢ ( 𝜑 → ( 𝐴 = ( 𝑀 ‘ 𝐴 ) ↔ ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐴 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝐴 𝐿 𝐴 ) ∨ 𝐴 = 𝐴 ) ) ) )
11		eqcom	⊢ ( 𝐴 = ( 𝑀 ‘ 𝐴 ) ↔ ( 𝑀 ‘ 𝐴 ) = 𝐴 )
12	11	a1i	⊢ ( 𝜑 → ( 𝐴 = ( 𝑀 ‘ 𝐴 ) ↔ ( 𝑀 ‘ 𝐴 ) = 𝐴 ) )
13		eqidd	⊢ ( 𝜑 → 𝐴 = 𝐴 )
14	13	olcd	⊢ ( 𝜑 → ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝐴 𝐿 𝐴 ) ∨ 𝐴 = 𝐴 ) )
15	14	biantrud	⊢ ( 𝜑 → ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐴 ) ∈ 𝐷 ↔ ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐴 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝐴 𝐿 𝐴 ) ∨ 𝐴 = 𝐴 ) ) ) )
16	1 2 3 4 5 9 9	midid	⊢ ( 𝜑 → ( 𝐴 ( midG ‘ 𝐺 ) 𝐴 ) = 𝐴 )
17	16	eleq1d	⊢ ( 𝜑 → ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐴 ) ∈ 𝐷 ↔ 𝐴 ∈ 𝐷 ) )
18	15 17	bitr3d	⊢ ( 𝜑 → ( ( ( 𝐴 ( midG ‘ 𝐺 ) 𝐴 ) ∈ 𝐷 ∧ ( 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝐴 𝐿 𝐴 ) ∨ 𝐴 = 𝐴 ) ) ↔ 𝐴 ∈ 𝐷 ) )
19	10 12 18	3bitr3d	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝐴 ) = 𝐴 ↔ 𝐴 ∈ 𝐷 ) )