Theorem miduniq1

Description: Uniqueness of the middle point, expressed with point inversion. Theorem 7.18 of Schwabhauser p. 52. (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	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	miduniq1.a	$$⊢\phi \to A\in P$$
	miduniq1.b	$$⊢\phi \to B\in P$$
	miduniq1.x	$$⊢\phi \to X\in P$$
	miduniq1.e	$$⊢\phi \to S\left(A\right)\left(X\right)=S\left(B\right)\left(X\right)$$
Assertion	miduniq1	$$⊢\phi \to A=B$$

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		miduniq1.a	$$⊢\phi \to A\in P$$
8		miduniq1.b	$$⊢\phi \to B\in P$$
9		miduniq1.x	$$⊢\phi \to X\in P$$
10		miduniq1.e	$$⊢\phi \to S\left(A\right)\left(X\right)=S\left(B\right)\left(X\right)$$
11		eqid	$$⊢S\left(A\right)=S\left(A\right)$$
12	1 2 3 4 5 6 7 11 9	mircl	$$⊢\phi \to S\left(A\right)\left(X\right)\in P$$
13		eqidd	$$⊢\phi \to S\left(A\right)\left(X\right)=S\left(A\right)\left(X\right)$$
14	10	eqcomd	$$⊢\phi \to S\left(B\right)\left(X\right)=S\left(A\right)\left(X\right)$$
15	1 2 3 4 5 6 7 8 9 12 13 14	miduniq	$$⊢\phi \to A=B$$