Theorem mircom

Description: Variation on mirmir . (Contributed by Thierry Arnoux, 10-Nov-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	$$⊢M=S\left(A\right)$$
	mirmir.b	$$⊢\phi \to B\in P$$
	mircom.1	$$⊢\phi \to M\left(B\right)=C$$
Assertion	mircom	$$⊢\phi \to M\left(C\right)=B$$

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		mirval.a	$$⊢\phi \to A\in P$$
8		mirfv.m	$$⊢M=S\left(A\right)$$
9		mirmir.b	$$⊢\phi \to B\in P$$
10		mircom.1	$$⊢\phi \to M\left(B\right)=C$$
11	10	fveq2d	$$⊢\phi \to M\left(M\left(B\right)\right)=M\left(C\right)$$
12	1 2 3 4 5 6 7 8 9	mirmir	$$⊢\phi \to M\left(M\left(B\right)\right)=B$$
13	11 12	eqtr3d	$$⊢\phi \to M\left(C\right)=B$$