Theorem mircgrs

Description: Point inversion preserves congruence. Theorem 7.16 of Schwabhauser p. 51. (Contributed by Thierry Arnoux, 30-Jul-2019)

	Ref	Expression
Hypotheses	mirval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	mirval.d	⊢ − = ( dist ‘ 𝐺 )
	mirval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	mirval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	mirval.s	⊢ 𝑆 = ( pInvG ‘ 𝐺 )
	mirval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	mirval.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	mirfv.m	⊢ 𝑀 = ( 𝑆 ‘ 𝐴 )
	miriso.1	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
	miriso.2	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
	mircgrs.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
	mircgrs.t	⊢ ( 𝜑 → 𝑇 ∈ 𝑃 )
	mircgrs.e	⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) = ( 𝑍 − 𝑇 ) )
Assertion	mircgrs	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) − ( 𝑀 ‘ 𝑌 ) ) = ( ( 𝑀 ‘ 𝑍 ) − ( 𝑀 ‘ 𝑇 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mirval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		mirval.d	⊢ − = ( dist ‘ 𝐺 )
3		mirval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		mirval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
5		mirval.s	⊢ 𝑆 = ( pInvG ‘ 𝐺 )
6		mirval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
7		mirval.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
8		mirfv.m	⊢ 𝑀 = ( 𝑆 ‘ 𝐴 )
9		miriso.1	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
10		miriso.2	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
11		mircgrs.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
12		mircgrs.t	⊢ ( 𝜑 → 𝑇 ∈ 𝑃 )
13		mircgrs.e	⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) = ( 𝑍 − 𝑇 ) )
14	1 2 3 4 5 6 7 8 9 10	miriso	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) − ( 𝑀 ‘ 𝑌 ) ) = ( 𝑋 − 𝑌 ) )
15	1 2 3 4 5 6 7 8 11 12	miriso	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑍 ) − ( 𝑀 ‘ 𝑇 ) ) = ( 𝑍 − 𝑇 ) )
16	13 14 15	3eqtr4d	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) − ( 𝑀 ‘ 𝑌 ) ) = ( ( 𝑀 ‘ 𝑍 ) − ( 𝑀 ‘ 𝑇 ) ) )