Theorem lmicinv

Description: The mirroring line is an invariant. (Contributed by Thierry Arnoux, 8-Aug-2020)

	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 )
	lmicinv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
Assertion	lmicinv	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) = 𝐴 )

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		lmicinv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
11	1 2 3 4 5 6 7 8 9	lmiinv	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝐴 ) = 𝐴 ↔ 𝐴 ∈ 𝐷 ) )
12	10 11	mpbird	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) = 𝐴 )