Theorem mirmir 28752

Description: The point inversion function is an involution. Theorem 7.7 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.)

Hypotheses

Ref	Expression
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)
mirval.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirfv.m	⊢ 𝑀 = (𝑆‘𝐴)
mirmir.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
mirmir	⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵)

Proof of Theorem mirmir

Step	Hyp	Ref	Expression
1		mirval.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		mirval.d	. . 3 ⊢ − = (dist‘𝐺)
3		mirval.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		mirval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
5		mirval.s	. . 3 ⊢ 𝑆 = (pInvG‘𝐺)
6		mirval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		mirval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8		mirfv.m	. . 3 ⊢ 𝑀 = (𝑆‘𝐴)
9		mirmir.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	mircl 28751	. . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃)
11	1, 2, 3, 4, 5, 6, 7, 8, 9	mircgr 28747	. . . 4 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵))
12	11	eqcomd 2743	. . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − (𝑀‘𝐵)))
13	1, 2, 3, 4, 5, 6, 7, 8, 9	mirbtwn 28748	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))
14	1, 2, 3, 6, 10, 7, 9, 13	tgbtwncom 28578	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼(𝑀‘𝐵)))
15	1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 12, 14	ismir 28749	. 2 ⊢ (𝜑 → 𝐵 = (𝑀‘(𝑀‘𝐵)))
16	15	eqcomd 2743	1 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6502  (class class class)co 7370  Basecbs 17150  distcds 17200  TarskiGcstrkg 28516  Itvcitv 28522  LineGclng 28523  pInvGcmir 28742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-trkgc 28537  df-trkgb 28538  df-trkgcb 28539  df-trkg 28542  df-mir 28743

This theorem is referenced by:  mircom  28753  mirreu  28754  mireq  28755  mirne  28757  mirf1o  28759  mirbtwnb  28762  miduniq2  28777  ragcom  28788  ragmir  28790  colperpexlem1  28820  colperpexlem2  28821  opphllem2  28838  opphllem3  28839  opphllem4  28840  opphllem6  28842  opphl  28844  colhp  28860  sacgr  28921