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Mirrors > Home > MPE Home > Th. List > mircl | Structured version Visualization version GIF version |
Description: Closure of the point inversion function. (Contributed by Thierry Arnoux, 20-Oct-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΄) |
mircl.x | β’ (π β π β π) |
Ref | Expression |
---|---|
mircl | β’ (π β (πβπ) β 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 | 1, 2, 3, 4, 5, 6, 7, 8 | mirf 28176 | . 2 β’ (π β π:πβΆπ) |
10 | mircl.x | . 2 β’ (π β π β π) | |
11 | 9, 10 | ffvelcdmd 7088 | 1 β’ (π β (πβπ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 βcfv 6544 Basecbs 17150 distcds 17212 TarskiGcstrkg 27943 Itvcitv 27949 LineGclng 27950 pInvGcmir 28168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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 7369 df-ov 7416 df-trkgc 27964 df-trkgb 27965 df-trkgcb 27966 df-trkg 27969 df-mir 28169 |
This theorem is referenced by: mirmir 28178 mirreu 28180 mireq 28181 miriso 28186 mirmir2 28190 mirln 28192 mirconn 28194 mirhl 28195 mirbtwnhl 28196 mirhl2 28197 mircgrextend 28198 mirtrcgr 28199 miduniq 28201 miduniq1 28202 miduniq2 28203 ragcom 28214 ragcol 28215 ragmir 28216 mirrag 28217 ragflat2 28219 ragflat 28220 ragcgr 28223 footexALT 28234 footexlem1 28235 footexlem2 28236 footex 28237 colperpexlem1 28246 colperpexlem3 28248 mideulem2 28250 opphllem 28251 opphllem2 28264 opphllem3 28265 opphllem4 28266 opphllem6 28268 opphl 28270 colhp 28286 mirmid 28299 lmieu 28300 lmimid 28310 lmiisolem 28312 hypcgrlem1 28315 hypcgrlem2 28316 hypcgr 28317 sacgr 28347 |
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