Theorem mirauto

Description: Point inversion preserves point inversion. (Contributed by Thierry Arnoux, 30-Jul-2019)

	Ref	Expression
Hypotheses	mirval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	mirval.d	⊢ − = ( dist ‘ 𝐺 )
	mirval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	mirval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	mirval.s	⊢ 𝑆 = ( pInvG ‘ 𝐺 )
	mirval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	mirauto.m	⊢ 𝑀 = ( 𝑆 ‘ 𝑇 )
	mirauto.x	⊢ 𝑋 = ( 𝑀 ‘ 𝐴 )
	mirauto.y	⊢ 𝑌 = ( 𝑀 ‘ 𝐵 )
	mirauto.z	⊢ 𝑍 = ( 𝑀 ‘ 𝐶 )
	mirauto.0	⊢ ( 𝜑 → 𝑇 ∈ 𝑃 )
	mirauto.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	mirauto.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	mirauto.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	mirauto.4	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐴 ) ‘ 𝐵 ) = 𝐶 )
Assertion	mirauto	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑋 ) ‘ 𝑌 ) = 𝑍 )

Proof

Step	Hyp	Ref	Expression
1		mirval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		mirval.d	⊢ − = ( dist ‘ 𝐺 )
3		mirval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		mirval.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
5		mirval.s	⊢ 𝑆 = ( pInvG ‘ 𝐺 )
6		mirval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
7		mirauto.m	⊢ 𝑀 = ( 𝑆 ‘ 𝑇 )
8		mirauto.x	⊢ 𝑋 = ( 𝑀 ‘ 𝐴 )
9		mirauto.y	⊢ 𝑌 = ( 𝑀 ‘ 𝐵 )
10		mirauto.z	⊢ 𝑍 = ( 𝑀 ‘ 𝐶 )
11		mirauto.0	⊢ ( 𝜑 → 𝑇 ∈ 𝑃 )
12		mirauto.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
13		mirauto.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
14		mirauto.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
15		mirauto.4	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐴 ) ‘ 𝐵 ) = 𝐶 )
16	1 2 3 4 5 6 11 7	mirf	⊢ ( 𝜑 → 𝑀 : 𝑃 ⟶ 𝑃 )
17	16 12	ffvelcdmd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ∈ 𝑃 )
18	8 17	eqeltrid	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
19		eqid	⊢ ( 𝑆 ‘ 𝑋 ) = ( 𝑆 ‘ 𝑋 )
20	16 13	ffvelcdmd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) ∈ 𝑃 )
21	9 20	eqeltrid	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
22	16 14	ffvelcdmd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐶 ) ∈ 𝑃 )
23	10 22	eqeltrid	⊢ ( 𝜑 → 𝑍 ∈ 𝑃 )
24	15 14	eqeltrd	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐴 ) ‘ 𝐵 ) ∈ 𝑃 )
25		eqid	⊢ ( 𝑆 ‘ 𝐴 ) = ( 𝑆 ‘ 𝐴 )
26	1 2 3 4 5 6 12 25 13	mircgr	⊢ ( 𝜑 → ( 𝐴 − ( ( 𝑆 ‘ 𝐴 ) ‘ 𝐵 ) ) = ( 𝐴 − 𝐵 ) )
27	1 2 3 4 5 6 11 7 12 24 12 13 26	mircgrs	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝐴 ) − ( 𝑀 ‘ ( ( 𝑆 ‘ 𝐴 ) ‘ 𝐵 ) ) ) = ( ( 𝑀 ‘ 𝐴 ) − ( 𝑀 ‘ 𝐵 ) ) )
28	8	a1i	⊢ ( 𝜑 → 𝑋 = ( 𝑀 ‘ 𝐴 ) )
29	15	fveq2d	⊢ ( 𝜑 → ( 𝑀 ‘ ( ( 𝑆 ‘ 𝐴 ) ‘ 𝐵 ) ) = ( 𝑀 ‘ 𝐶 ) )
30	10 29	eqtr4id	⊢ ( 𝜑 → 𝑍 = ( 𝑀 ‘ ( ( 𝑆 ‘ 𝐴 ) ‘ 𝐵 ) ) )
31	28 30	oveq12d	⊢ ( 𝜑 → ( 𝑋 − 𝑍 ) = ( ( 𝑀 ‘ 𝐴 ) − ( 𝑀 ‘ ( ( 𝑆 ‘ 𝐴 ) ‘ 𝐵 ) ) ) )
32	8 9	oveq12i	⊢ ( 𝑋 − 𝑌 ) = ( ( 𝑀 ‘ 𝐴 ) − ( 𝑀 ‘ 𝐵 ) )
33	32	a1i	⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) = ( ( 𝑀 ‘ 𝐴 ) − ( 𝑀 ‘ 𝐵 ) ) )
34	27 31 33	3eqtr4d	⊢ ( 𝜑 → ( 𝑋 − 𝑍 ) = ( 𝑋 − 𝑌 ) )
35	1 2 3 4 5 6 12 25 13	mirbtwn	⊢ ( 𝜑 → 𝐴 ∈ ( ( ( 𝑆 ‘ 𝐴 ) ‘ 𝐵 ) 𝐼 𝐵 ) )
36	15	oveq1d	⊢ ( 𝜑 → ( ( ( 𝑆 ‘ 𝐴 ) ‘ 𝐵 ) 𝐼 𝐵 ) = ( 𝐶 𝐼 𝐵 ) )
37	35 36	eleqtrd	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) )
38	1 2 3 4 5 6 11 7 14 12 13 37	mirbtwni	⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ∈ ( ( 𝑀 ‘ 𝐶 ) 𝐼 ( 𝑀 ‘ 𝐵 ) ) )
39	10 9	oveq12i	⊢ ( 𝑍 𝐼 𝑌 ) = ( ( 𝑀 ‘ 𝐶 ) 𝐼 ( 𝑀 ‘ 𝐵 ) )
40	38 8 39	3eltr4g	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) )
41	1 2 3 4 5 6 18 19 21 23 34 40	ismir	⊢ ( 𝜑 → 𝑍 = ( ( 𝑆 ‘ 𝑋 ) ‘ 𝑌 ) )
42	41	eqcomd	⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑋 ) ‘ 𝑌 ) = 𝑍 )