Theorem mirinv 28801

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 483	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐺 ∈ TarskiG)
6		mirinv.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7	6	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐵 ∈ 𝑃)
8		mirval.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	8	adantr 483	. . . 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 28793	. . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))
14		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → (𝑀‘𝐵) = 𝐵)
15	14	oveq1d 7396	. . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → ((𝑀‘𝐵)𝐼𝐵) = (𝐵𝐼𝐵))
16	13, 15	eleqtrd 2854	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
17	1, 2, 3, 5, 7, 9, 16	axtgbtwnid 28601	. . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐵 = 𝐴)
18	17	eqcomd 2758	. 2 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 = 𝐵)
19	4	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
20	8	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃)
21	6	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃)
22		eqidd 2753	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐴 − 𝐵))
23		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
24	1, 2, 3, 19, 21, 21	tgbtwntriv1 28626	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ (𝐵𝐼𝐵))
25	23, 24	eqeltrd 2852	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
26	1, 2, 3, 10, 11, 19, 20, 12, 21, 21, 22, 25	ismir 28794	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 = (𝑀‘𝐵))
27	26	eqcomd 2758	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) = 𝐵)
28	18, 27	impbida 808	1 ⊢ (𝜑 → ((𝑀‘𝐵) = 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1550   ∈ wcel 2132  ‘cfv 6506  (class class class)co 7381  Basecbs 17217  distcds 17267  TarskiGcstrkg 28562  Itvcitv 28568  LineGclng 28569  pInvGcmir 28787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-trkgc 28583  df-trkgb 28584  df-trkgcb 28585  df-trkg 28588  df-mir 28788

This theorem is referenced by:  mirne  28802  mircinv  28803  mirln2  28812  miduniq  28820  miduniq2  28822  krippenlem  28825  ragflat2  28838  footexALT  28853  footexlem1  28854  footexlem2  28855  colperpexlem2  28866  colperpexlem3  28867  opphllem6  28887  lmimid  28929  hypcgrlem2  28935