Theorem ismir

Description: Property of the image by the point inversion function. Definition 7.5 of Schwabhauser p. 49. (Contributed by Thierry Arnoux, 3-Jun-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	$$⊢M=S\left(A\right)$$
	mirfv.b	$$⊢\phi \to B\in P$$
	ismir.1	$$⊢\phi \to C\in P$$
	ismir.2	$$⊢\phi \to A\underset{\dot{}}{-}C=A\underset{\dot{}}{-}B$$
	ismir.3	$$⊢\phi \to A\in \left(CIB\right)$$
Assertion	ismir	$$⊢\phi \to C=M\left(B\right)$$

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		mirfv.b	$$⊢\phi \to B\in P$$
10		ismir.1	$$⊢\phi \to C\in P$$
11		ismir.2	$$⊢\phi \to A\underset{\dot{}}{-}C=A\underset{\dot{}}{-}B$$
12		ismir.3	$$⊢\phi \to A\in \left(CIB\right)$$
13	1 2 3 4 5 6 7 8 9	mirfv	$$⊢\phi \to M\left(B\right)=\left(\iota z\in P|\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\right)$$
14	1 2 3 6 9 7	mirreu3	$$⊢\phi \to \exists !z\in P\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)$$
15		oveq2	$$⊢z=C\to A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}C$$
16	15	eqeq1d	$$⊢z=C\to \left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\leftrightarrow A\underset{\dot{}}{-}C=A\underset{\dot{}}{-}B\right)$$
17		oveq1	$$⊢z=C\to zIB=CIB$$
18	17	eleq2d	$$⊢z=C\to \left(A\in \left(zIB\right)\leftrightarrow A\in \left(CIB\right)\right)$$
19	16 18	anbi12d	$$⊢z=C\to \left(\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\leftrightarrow \left(A\underset{\dot{}}{-}C=A\underset{\dot{}}{-}B\wedge A\in \left(CIB\right)\right)\right)$$
20	19	riota2	$$⊢\left(C\in P\wedge \exists !z\in P\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\right)\to \left(\left(A\underset{\dot{}}{-}C=A\underset{\dot{}}{-}B\wedge A\in \left(CIB\right)\right)\leftrightarrow \left(\iota z\in P|\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\right)=C\right)$$
21	10 14 20	syl2anc	$$⊢\phi \to \left(\left(A\underset{\dot{}}{-}C=A\underset{\dot{}}{-}B\wedge A\in \left(CIB\right)\right)\leftrightarrow \left(\iota z\in P|\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\right)=C\right)$$
22	11 12 21	mpbi2and	$$⊢\phi \to \left(\iota z\in P|\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\right)=C$$
23	13 22	eqtr2d	$$⊢\phi \to C=M\left(B\right)$$