Metamath Proof Explorer

Theorem lmiiso

Description: The line mirroring function is an isometry, i.e. it is conserves congruence. Because it is also a bijection, it is also a motion. Theorem 10.10 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 Dim_TarskiG≥ 2 )
		lmif.m	⊢ 𝑀 = ( ( lInvG ‘ 𝐺 ) ‘ 𝐷 )
		lmif.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
		lmif.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
		lmiiso.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
		lmiiso.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	Assertion	lmiiso	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝐴 ) − ( 𝑀 ‘ 𝐵 ) ) = ( 𝐴 − 𝐵 ) )

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 Dim_TarskiG≥ 2 )
6		lmif.m	⊢ 𝑀 = ( ( lInvG ‘ 𝐺 ) ‘ 𝐷 )
7		lmif.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
8		lmif.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
9		lmiiso.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
10		lmiiso.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
11		eqid	⊢ ( ( pInvG ‘ 𝐺 ) ‘ ( ( 𝐴 ( midG ‘ 𝐺 ) ( 𝑀 ‘ 𝐴 ) ) ( midG ‘ 𝐺 ) ( 𝐵 ( midG ‘ 𝐺 ) ( 𝑀 ‘ 𝐵 ) ) ) ) = ( ( pInvG ‘ 𝐺 ) ‘ ( ( 𝐴 ( midG ‘ 𝐺 ) ( 𝑀 ‘ 𝐴 ) ) ( midG ‘ 𝐺 ) ( 𝐵 ( midG ‘ 𝐺 ) ( 𝑀 ‘ 𝐵 ) ) ) )
12		eqid	⊢ ( ( 𝐴 ( midG ‘ 𝐺 ) ( 𝑀 ‘ 𝐴 ) ) ( midG ‘ 𝐺 ) ( 𝐵 ( midG ‘ 𝐺 ) ( 𝑀 ‘ 𝐵 ) ) ) = ( ( 𝐴 ( midG ‘ 𝐺 ) ( 𝑀 ‘ 𝐴 ) ) ( midG ‘ 𝐺 ) ( 𝐵 ( midG ‘ 𝐺 ) ( 𝑀 ‘ 𝐵 ) ) )
13	1 2 3 4 5 6 7 8 9 10 11 12	lmiisolem	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝐴 ) − ( 𝑀 ‘ 𝐵 ) ) = ( 𝐴 − 𝐵 ) )