Theorem mirinv 28893

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 485	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐺 ∈ TarskiG)
6		mirinv.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7	6	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐵 ∈ 𝑃)
8		mirval.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	8	adantr 485	. . . 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 28885	. . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))
14		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → (𝑀‘𝐵) = 𝐵)
15	14	oveq1d 7415	. . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → ((𝑀‘𝐵)𝐼𝐵) = (𝐵𝐼𝐵))
16	13, 15	eleqtrd 2867	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
17	1, 2, 3, 5, 7, 9, 16	axtgbtwnid 28689	. . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐵 = 𝐴)
18	17	eqcomd 2771	. 2 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 = 𝐵)
19	4	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
20	8	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃)
21	6	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃)
22		eqidd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐴 − 𝐵))
23		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
24	1, 2, 3, 19, 21, 21	tgbtwntriv1 28714	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ (𝐵𝐼𝐵))
25	23, 24	eqeltrd 2865	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
26	1, 2, 3, 10, 11, 19, 20, 12, 21, 21, 22, 25	ismir 28886	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 = (𝑀‘𝐵))
27	26	eqcomd 2771	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) = 𝐵)
28	18, 27	impbida 812	1 ⊢ (𝜑 → ((𝑀‘𝐵) = 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ‘cfv 6525  (class class class)co 7400  Basecbs 17257  distcds 17307  TarskiGcstrkg 28650  Itvcitv 28656  LineGclng 28657  pInvGcmir 28879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-trkgc 28671  df-trkgb 28672  df-trkgcb 28673  df-trkg 28676  df-mir 28880

This theorem is referenced by:  mirne  28894  mircinv  28895  mirln2  28904  miduniq  28912  miduniq2  28914  krippenlem  28917  ragflat2  28930  footexALT  28945  footexlem1  28946  footexlem2  28947  colperpexlem2  28958  colperpexlem3  28959  opphllem6  28979  lmimid  29042  hypcgrlem2  29048