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	$$⊢P={\mathrm{Base}}_G$$
	mirval.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	mirval.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	mirval.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
	mirval.s	$$⊢S={pInv}_{𝒢}\left(G\right)$$
	mirval.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	mirval.a	$$⊢\phi \to A\in P$$
	mirfv.m	$$⊢M=S\left(A\right)$$
	miriso.1	$$⊢\phi \to X\in P$$
	miriso.2	$$⊢\phi \to Y\in P$$
	mircgrs.z	$$⊢\phi \to Z\in P$$
	mircgrs.t	$$⊢\phi \to T\in P$$
	mircgrs.e	$$⊢\phi \to X\underset{\dot{}}{-}Y=Z\underset{\dot{}}{-}T$$
Assertion	mircgrs	$$⊢\phi \to M\left(X\right)\underset{\dot{}}{-}M\left(Y\right)=M\left(Z\right)\underset{\dot{}}{-}M\left(T\right)$$

Proof

Step	Hyp	Ref	Expression
1		mirval.p	$$⊢P={\mathrm{Base}}_G$$
2		mirval.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		mirval.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		mirval.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
5		mirval.s	$$⊢S={pInv}_{𝒢}\left(G\right)$$
6		mirval.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
7		mirval.a	$$⊢\phi \to A\in P$$
8		mirfv.m	$$⊢M=S\left(A\right)$$
9		miriso.1	$$⊢\phi \to X\in P$$
10		miriso.2	$$⊢\phi \to Y\in P$$
11		mircgrs.z	$$⊢\phi \to Z\in P$$
12		mircgrs.t	$$⊢\phi \to T\in P$$
13		mircgrs.e	$$⊢\phi \to X\underset{\dot{}}{-}Y=Z\underset{\dot{}}{-}T$$
14	1 2 3 4 5 6 7 8 9 10	miriso	$$⊢\phi \to M\left(X\right)\underset{\dot{}}{-}M\left(Y\right)=X\underset{\dot{}}{-}Y$$
15	1 2 3 4 5 6 7 8 11 12	miriso	$$⊢\phi \to M\left(Z\right)\underset{\dot{}}{-}M\left(T\right)=Z\underset{\dot{}}{-}T$$
16	13 14 15	3eqtr4d	$$⊢\phi \to M\left(X\right)\underset{\dot{}}{-}M\left(Y\right)=M\left(Z\right)\underset{\dot{}}{-}M\left(T\right)$$