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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 27907 | . . . 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 27905 | . . . . 5 β’ (π β β!π§ β π ((π΄ β π§) = (π΄ β π΅) β§ π΄ β (π§πΌπ΅))) |
12 | riotacl2 7382 | . . . . 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 2834 | . . 3 β’ (π β (πβπ΅) β {π§ β π β£ ((π΄ β π§) = (π΄ β π΅) β§ π΄ β (π§πΌπ΅))}) |
15 | oveq2 7417 | . . . . . 6 β’ (π§ = (πβπ΅) β (π΄ β π§) = (π΄ β (πβπ΅))) | |
16 | 15 | eqeq1d 2735 | . . . . 5 β’ (π§ = (πβπ΅) β ((π΄ β π§) = (π΄ β π΅) β (π΄ β (πβπ΅)) = (π΄ β π΅))) |
17 | oveq1 7416 | . . . . . 6 β’ (π§ = (πβπ΅) β (π§πΌπ΅) = ((πβπ΅)πΌπ΅)) | |
18 | 17 | eleq2d 2820 | . . . . 5 β’ (π§ = (πβπ΅) β (π΄ β (π§πΌπ΅) β π΄ β ((πβπ΅)πΌπ΅))) |
19 | 16, 18 | anbi12d 632 | . . . 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 3684 | . . 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 773 | 1 β’ (π β π΄ β ((πβπ΅)πΌπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β!wreu 3375 {crab 3433 βcfv 6544 β©crio 7364 (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: mirmir 27913 mirinv 27917 miriso 27921 mirmir2 27925 mirln 27927 mirln2 27928 mirconn 27929 mirhl2 27932 mircgrextend 27933 mirtrcgr 27934 mirauto 27935 miduniq 27936 krippenlem 27941 ragflat 27955 ragcgr 27958 footexALT 27969 footexlem1 27970 footexlem2 27971 colperpexlem1 27981 colperpexlem3 27983 mideulem2 27985 opphllem 27986 opphllem1 27998 opphllem2 27999 opphllem4 28001 colhp 28021 midbtwn 28030 lmieu 28035 lmiisolem 28047 |
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