Theorem midcom

Description: Commutativity rule for the midpoint. (Contributed by Thierry Arnoux, 2-Dec-2019)

	Ref	Expression
Hypotheses	ismid.p	$$⊢P={\mathrm{Base}}_G$$
	ismid.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	ismid.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	ismid.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	ismid.1	$$⊢\phi \to G{\mathrm{Dim}}_{𝒢}\ge 2$$
	midcl.1	$$⊢\phi \to A\in P$$
	midcl.2	$$⊢\phi \to B\in P$$
Assertion	midcom	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B=B{mid}_{𝒢}\left(G\right)A$$

Proof

Step	Hyp	Ref	Expression
1		ismid.p	$$⊢P={\mathrm{Base}}_G$$
2		ismid.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		ismid.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		ismid.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		ismid.1	$$⊢\phi \to G{\mathrm{Dim}}_{𝒢}\ge 2$$
6		midcl.1	$$⊢\phi \to A\in P$$
7		midcl.2	$$⊢\phi \to B\in P$$
8		eqid	$$⊢{Line}_{𝒢}\left(G\right)={Line}_{𝒢}\left(G\right)$$
9		eqid	$$⊢{pInv}_{𝒢}\left(G\right)={pInv}_{𝒢}\left(G\right)$$
10	1 2 3 4 5 7 6	midcl	$$⊢\phi \to B{mid}_{𝒢}\left(G\right)A\in P$$
11		eqid	$$⊢{pInv}_{𝒢}\left(G\right)\left(B{mid}_{𝒢}\left(G\right)A\right)={pInv}_{𝒢}\left(G\right)\left(B{mid}_{𝒢}\left(G\right)A\right)$$
12		eqidd	$$⊢\phi \to B{mid}_{𝒢}\left(G\right)A=B{mid}_{𝒢}\left(G\right)A$$
13	1 2 3 4 5 7 6 12	midcgr	$$⊢\phi \to \left(B{mid}_{𝒢}\left(G\right)A\right)\underset{\dot{}}{-}B=\left(B{mid}_{𝒢}\left(G\right)A\right)\underset{\dot{}}{-}A$$
14	1 2 3 4 5 7 6	midbtwn	$$⊢\phi \to B{mid}_{𝒢}\left(G\right)A\in \left(BIA\right)$$
15	1 2 3 8 9 4 10 11 6 7 13 14	ismir	$$⊢\phi \to B={pInv}_{𝒢}\left(G\right)\left(B{mid}_{𝒢}\left(G\right)A\right)\left(A\right)$$
16	1 2 3 4 5 6 7 9 10	ismidb	$$⊢\phi \to \left(B={pInv}_{𝒢}\left(G\right)\left(B{mid}_{𝒢}\left(G\right)A\right)\left(A\right)\leftrightarrow A{mid}_{𝒢}\left(G\right)B=B{mid}_{𝒢}\left(G\right)A\right)$$
17	15 16	mpbid	$$⊢\phi \to A{mid}_{𝒢}\left(G\right)B=B{mid}_{𝒢}\left(G\right)A$$