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Mirrors > Home > MPE Home > Th. List > mirmir | Structured version Visualization version GIF version |
Description: The point inversion function is an involution. Theorem 7.7 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.) |
Ref | Expression |
---|---|
mirval.p | β’ π = (BaseβπΊ) |
mirval.d | β’ β = (distβπΊ) |
mirval.i | β’ πΌ = (ItvβπΊ) |
mirval.l | β’ πΏ = (LineGβπΊ) |
mirval.s | β’ π = (pInvGβπΊ) |
mirval.g | β’ (π β πΊ β TarskiG) |
mirval.a | β’ (π β π΄ β π) |
mirfv.m | β’ π = (πβπ΄) |
mirmir.b | β’ (π β π΅ β π) |
Ref | Expression |
---|---|
mirmir | β’ (π β (πβ(πβπ΅)) = π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . . 3 β’ π = (BaseβπΊ) | |
2 | mirval.d | . . 3 β’ β = (distβπΊ) | |
3 | mirval.i | . . 3 β’ πΌ = (ItvβπΊ) | |
4 | mirval.l | . . 3 β’ πΏ = (LineGβπΊ) | |
5 | mirval.s | . . 3 β’ π = (pInvGβπΊ) | |
6 | mirval.g | . . 3 β’ (π β πΊ β TarskiG) | |
7 | mirval.a | . . 3 β’ (π β π΄ β π) | |
8 | mirfv.m | . . 3 β’ π = (πβπ΄) | |
9 | mirmir.b | . . . 4 β’ (π β π΅ β π) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mircl 27912 | . . 3 β’ (π β (πβπ΅) β π) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mircgr 27908 | . . . 4 β’ (π β (π΄ β (πβπ΅)) = (π΄ β π΅)) |
12 | 11 | eqcomd 2739 | . . 3 β’ (π β (π΄ β π΅) = (π΄ β (πβπ΅))) |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mirbtwn 27909 | . . . 4 β’ (π β π΄ β ((πβπ΅)πΌπ΅)) |
14 | 1, 2, 3, 6, 10, 7, 9, 13 | tgbtwncom 27739 | . . 3 β’ (π β π΄ β (π΅πΌ(πβπ΅))) |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 12, 14 | ismir 27910 | . 2 β’ (π β π΅ = (πβ(πβπ΅))) |
16 | 15 | eqcomd 2739 | 1 β’ (π β (πβ(πβπ΅)) = π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6544 (class class class)co 7409 Basecbs 17144 distcds 17206 TarskiGcstrkg 27678 Itvcitv 27684 LineGclng 27685 pInvGcmir 27903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-trkgc 27699 df-trkgb 27700 df-trkgcb 27701 df-trkg 27704 df-mir 27904 |
This theorem is referenced by: mircom 27914 mirreu 27915 mireq 27916 mirne 27918 mirf1o 27920 mirbtwnb 27923 miduniq2 27938 ragcom 27949 ragmir 27951 colperpexlem1 27981 colperpexlem2 27982 opphllem2 27999 opphllem3 28000 opphllem4 28001 opphllem6 28003 opphl 28005 colhp 28021 sacgr 28082 |
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