Theorem mirinv 26931

Description: The only invariant point of a point inversion Theorem 7.3 of [Schwabhauser] p. 49, Theorem 7.10 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 30-Jul-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)
mirinv.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
mirinv	⊢ (𝜑 → ((𝑀‘𝐵) = 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem mirinv

Step	Hyp	Ref	Expression
1		mirval.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		mirval.d	. . . 4 ⊢ − = (dist‘𝐺)
3		mirval.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		mirval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐺 ∈ TarskiG)
6		mirinv.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐵 ∈ 𝑃)
8		mirval.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 ∈ 𝑃)
10		mirval.l	. . . . . 6 ⊢ 𝐿 = (LineG‘𝐺)
11		mirval.s	. . . . . 6 ⊢ 𝑆 = (pInvG‘𝐺)
12		mirfv.m	. . . . . 6 ⊢ 𝑀 = (𝑆‘𝐴)
13	1, 2, 3, 10, 11, 5, 9, 12, 7	mirbtwn 26923	. . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))
14		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → (𝑀‘𝐵) = 𝐵)
15	14	oveq1d 7270	. . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → ((𝑀‘𝐵)𝐼𝐵) = (𝐵𝐼𝐵))
16	13, 15	eleqtrd 2841	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
17	1, 2, 3, 5, 7, 9, 16	axtgbtwnid 26731	. . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐵 = 𝐴)
18	17	eqcomd 2744	. 2 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 = 𝐵)
19	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
20	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃)
21	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃)
22		eqidd 2739	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐴 − 𝐵))
23		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
24	1, 2, 3, 19, 21, 21	tgbtwntriv1 26756	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ (𝐵𝐼𝐵))
25	23, 24	eqeltrd 2839	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
26	1, 2, 3, 10, 11, 19, 20, 12, 21, 21, 22, 25	ismir 26924	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 = (𝑀‘𝐵))
27	26	eqcomd 2744	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) = 𝐵)
28	18, 27	impbida 797	1 ⊢ (𝜑 → ((𝑀‘𝐵) = 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  distcds 16897  TarskiGcstrkg 26693  Itvcitv 26699  LineGclng 26700  pInvGcmir 26917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-trkgc 26713  df-trkgb 26714  df-trkgcb 26715  df-trkg 26718  df-mir 26918

This theorem is referenced by:  mirne  26932  mircinv  26933  mirln2  26942  miduniq  26950  miduniq2  26952  krippenlem  26955  ragflat2  26968  footexALT  26983  footexlem1  26984  footexlem2  26985  colperpexlem2  26996  colperpexlem3  26997  opphllem6  27017  lmimid  27059  hypcgrlem2  27065