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Mirrors > Home > MPE Home > Th. List > mirbtwn | Structured version Visualization version GIF version |
Description: Property of the image by the point inversion function. Definition 7.5 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΄) |
mirfv.b | β’ (π β π΅ β π) |
Ref | Expression |
---|---|
mirbtwn | β’ (π β π΄ β ((πβπ΅)πΌπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . . . . 5 β’ π = (BaseβπΊ) | |
2 | mirval.d | . . . . 5 β’ β = (distβπΊ) | |
3 | mirval.i | . . . . 5 β’ πΌ = (ItvβπΊ) | |
4 | mirval.l | . . . . 5 β’ πΏ = (LineGβπΊ) | |
5 | mirval.s | . . . . 5 β’ π = (pInvGβπΊ) | |
6 | mirval.g | . . . . 5 β’ (π β πΊ β TarskiG) | |
7 | mirval.a | . . . . 5 β’ (π β π΄ β π) | |
8 | mirfv.m | . . . . 5 β’ π = (πβπ΄) | |
9 | mirfv.b | . . . . 5 β’ (π β π΅ β π) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mirfv 28162 | . . . 4 β’ (π β (πβπ΅) = (β©π§ β π ((π΄ β π§) = (π΄ β π΅) β§ π΄ β (π§πΌπ΅)))) |
11 | 1, 2, 3, 6, 9, 7 | mirreu3 28160 | . . . . 5 β’ (π β β!π§ β π ((π΄ β π§) = (π΄ β π΅) β§ π΄ β (π§πΌπ΅))) |
12 | riotacl2 7384 | . . . . 5 β’ (β!π§ β π ((π΄ β π§) = (π΄ β π΅) β§ π΄ β (π§πΌπ΅)) β (β©π§ β π ((π΄ β π§) = (π΄ β π΅) β§ π΄ β (π§πΌπ΅))) β {π§ β π β£ ((π΄ β π§) = (π΄ β π΅) β§ π΄ β (π§πΌπ΅))}) | |
13 | 11, 12 | syl 17 | . . . 4 β’ (π β (β©π§ β π ((π΄ β π§) = (π΄ β π΅) β§ π΄ β (π§πΌπ΅))) β {π§ β π β£ ((π΄ β π§) = (π΄ β π΅) β§ π΄ β (π§πΌπ΅))}) |
14 | 10, 13 | eqeltrd 2833 | . . 3 β’ (π β (πβπ΅) β {π§ β π β£ ((π΄ β π§) = (π΄ β π΅) β§ π΄ β (π§πΌπ΅))}) |
15 | oveq2 7419 | . . . . . 6 β’ (π§ = (πβπ΅) β (π΄ β π§) = (π΄ β (πβπ΅))) | |
16 | 15 | eqeq1d 2734 | . . . . 5 β’ (π§ = (πβπ΅) β ((π΄ β π§) = (π΄ β π΅) β (π΄ β (πβπ΅)) = (π΄ β π΅))) |
17 | oveq1 7418 | . . . . . 6 β’ (π§ = (πβπ΅) β (π§πΌπ΅) = ((πβπ΅)πΌπ΅)) | |
18 | 17 | eleq2d 2819 | . . . . 5 β’ (π§ = (πβπ΅) β (π΄ β (π§πΌπ΅) β π΄ β ((πβπ΅)πΌπ΅))) |
19 | 16, 18 | anbi12d 631 | . . . 4 β’ (π§ = (πβπ΅) β (((π΄ β π§) = (π΄ β π΅) β§ π΄ β (π§πΌπ΅)) β ((π΄ β (πβπ΅)) = (π΄ β π΅) β§ π΄ β ((πβπ΅)πΌπ΅)))) |
20 | 19 | elrab 3683 | . . 3 β’ ((πβπ΅) β {π§ β π β£ ((π΄ β π§) = (π΄ β π΅) β§ π΄ β (π§πΌπ΅))} β ((πβπ΅) β π β§ ((π΄ β (πβπ΅)) = (π΄ β π΅) β§ π΄ β ((πβπ΅)πΌπ΅)))) |
21 | 14, 20 | sylib 217 | . 2 β’ (π β ((πβπ΅) β π β§ ((π΄ β (πβπ΅)) = (π΄ β π΅) β§ π΄ β ((πβπ΅)πΌπ΅)))) |
22 | 21 | simprrd 772 | 1 β’ (π β π΄ β ((πβπ΅)πΌπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β!wreu 3374 {crab 3432 βcfv 6543 β©crio 7366 (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: mirmir 28168 mirinv 28172 miriso 28176 mirmir2 28180 mirln 28182 mirln2 28183 mirconn 28184 mirhl2 28187 mircgrextend 28188 mirtrcgr 28189 mirauto 28190 miduniq 28191 krippenlem 28196 ragflat 28210 ragcgr 28213 footexALT 28224 footexlem1 28225 footexlem2 28226 colperpexlem1 28236 colperpexlem3 28238 mideulem2 28240 opphllem 28241 opphllem1 28253 opphllem2 28254 opphllem4 28256 colhp 28276 midbtwn 28285 lmieu 28290 lmiisolem 28302 |
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