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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 28167 | . . 3 β’ (π β (πβπ΅) β π) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mircgr 28163 | . . . 4 β’ (π β (π΄ β (πβπ΅)) = (π΄ β π΅)) |
12 | 11 | eqcomd 2738 | . . 3 β’ (π β (π΄ β π΅) = (π΄ β (πβπ΅))) |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mirbtwn 28164 | . . . 4 β’ (π β π΄ β ((πβπ΅)πΌπ΅)) |
14 | 1, 2, 3, 6, 10, 7, 9, 13 | tgbtwncom 27994 | . . 3 β’ (π β π΄ β (π΅πΌ(πβπ΅))) |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 12, 14 | ismir 28165 | . 2 β’ (π β π΅ = (πβ(πβπ΅))) |
16 | 15 | eqcomd 2738 | 1 β’ (π β (πβ(πβπ΅)) = π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 βcfv 6543 (class class class)co 7411 Basecbs 17148 distcds 17210 TarskiGcstrkg 27933 Itvcitv 27939 LineGclng 27940 pInvGcmir 28158 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-trkgc 27954 df-trkgb 27955 df-trkgcb 27956 df-trkg 27959 df-mir 28159 |
This theorem is referenced by: mircom 28169 mirreu 28170 mireq 28171 mirne 28173 mirf1o 28175 mirbtwnb 28178 miduniq2 28193 ragcom 28204 ragmir 28206 colperpexlem1 28236 colperpexlem2 28237 opphllem2 28254 opphllem3 28255 opphllem4 28256 opphllem6 28258 opphl 28260 colhp 28276 sacgr 28337 |
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