Theorem mirbtwn

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	$$⊢\phi \to B\in P$$
Assertion	mirbtwn	$$⊢\phi \to A\in \left(M\left(B\right)IB\right)$$

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	1 2 3 4 5 6 7 8 9	mirfv	$$⊢\phi \to M\left(B\right)=\left(\iota z\in P|\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\right)$$
11	1 2 3 6 9 7	mirreu3	$$⊢\phi \to \exists !z\in P\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)$$
12		riotacl2	$$⊢\exists !z\in P\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\to \left(\iota z\in P|\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\right)\in \left\{z\in P|\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\right\}$$
13	11 12	syl	$$⊢\phi \to \left(\iota z\in P|\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\right)\in \left\{z\in P|\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\right\}$$
14	10 13	eqeltrd	$$⊢\phi \to M\left(B\right)\in \left\{z\in P|\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\right\}$$
15		oveq2	$$⊢z=M\left(B\right)\to A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}M\left(B\right)$$
16	15	eqeq1d	$$⊢z=M\left(B\right)\to \left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\leftrightarrow A\underset{\dot{}}{-}M\left(B\right)=A\underset{\dot{}}{-}B\right)$$
17		oveq1	$$⊢z=M\left(B\right)\to zIB=M\left(B\right)IB$$
18	17	eleq2d	$$⊢z=M\left(B\right)\to \left(A\in \left(zIB\right)\leftrightarrow A\in \left(M\left(B\right)IB\right)\right)$$
19	16 18	anbi12d	$$⊢z=M\left(B\right)\to \left(\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\leftrightarrow \left(A\underset{\dot{}}{-}M\left(B\right)=A\underset{\dot{}}{-}B\wedge A\in \left(M\left(B\right)IB\right)\right)\right)$$
20	19	elrab	$$⊢M\left(B\right)\in \left\{z\in P|\left(A\underset{\dot{}}{-}z=A\underset{\dot{}}{-}B\wedge A\in \left(zIB\right)\right)\right\}\leftrightarrow \left(M\left(B\right)\in P\wedge \left(A\underset{\dot{}}{-}M\left(B\right)=A\underset{\dot{}}{-}B\wedge A\in \left(M\left(B\right)IB\right)\right)\right)$$
21	14 20	sylib	$$⊢\phi \to \left(M\left(B\right)\in P\wedge \left(A\underset{\dot{}}{-}M\left(B\right)=A\underset{\dot{}}{-}B\wedge A\in \left(M\left(B\right)IB\right)\right)\right)$$
22	21	simprrd	$$⊢\phi \to A\in \left(M\left(B\right)IB\right)$$