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Mirrors > Home > MPE Home > Th. List > mirhl2 | Structured version Visualization version GIF version |
Description: Deduce half-line relation from mirror point. (Contributed by Thierry Arnoux, 8-Aug-2020.) |
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) |
mirhl.m | β’ π = (πβπ΄) |
mirhl.k | β’ πΎ = (hlGβπΊ) |
mirhl.a | β’ (π β π΄ β π) |
mirhl.x | β’ (π β π β π) |
mirhl.y | β’ (π β π β π) |
mirhl.z | β’ (π β π β π) |
mirhl2.1 | β’ (π β π β π΄) |
mirhl2.2 | β’ (π β π β π΄) |
mirhl2.3 | β’ (π β π΄ β (ππΌ(πβπ))) |
Ref | Expression |
---|---|
mirhl2 | β’ (π β π(πΎβπ΄)π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirhl2.1 | . 2 β’ (π β π β π΄) | |
2 | mirhl2.2 | . 2 β’ (π β π β π΄) | |
3 | mirval.p | . . 3 β’ π = (BaseβπΊ) | |
4 | mirval.i | . . 3 β’ πΌ = (ItvβπΊ) | |
5 | mirval.g | . . 3 β’ (π β πΊ β TarskiG) | |
6 | mirval.d | . . . 4 β’ β = (distβπΊ) | |
7 | mirval.l | . . . 4 β’ πΏ = (LineGβπΊ) | |
8 | mirval.s | . . . 4 β’ π = (pInvGβπΊ) | |
9 | mirhl.a | . . . 4 β’ (π β π΄ β π) | |
10 | mirhl.m | . . . 4 β’ π = (πβπ΄) | |
11 | mirhl.y | . . . 4 β’ (π β π β π) | |
12 | 3, 6, 4, 7, 8, 5, 9, 10, 11 | mircl 28478 | . . 3 β’ (π β (πβπ) β π) |
13 | mirhl.x | . . 3 β’ (π β π β π) | |
14 | 3, 6, 4, 7, 8, 5, 9, 10, 11, 2 | mirne 28484 | . . 3 β’ (π β (πβπ) β π΄) |
15 | mirhl2.3 | . . . 4 β’ (π β π΄ β (ππΌ(πβπ))) | |
16 | 3, 6, 4, 5, 13, 9, 12, 15 | tgbtwncom 28305 | . . 3 β’ (π β π΄ β ((πβπ)πΌπ)) |
17 | 3, 6, 4, 7, 8, 5, 9, 10, 11 | mirbtwn 28475 | . . 3 β’ (π β π΄ β ((πβπ)πΌπ)) |
18 | 3, 4, 5, 12, 9, 13, 11, 14, 16, 17 | tgbtwnconn2 28393 | . 2 β’ (π β (π β (π΄πΌπ) β¨ π β (π΄πΌπ))) |
19 | mirhl.k | . . 3 β’ πΎ = (hlGβπΊ) | |
20 | 3, 4, 19, 13, 11, 9, 5 | ishlg 28419 | . 2 β’ (π β (π(πΎβπ΄)π β (π β π΄ β§ π β π΄ β§ (π β (π΄πΌπ) β¨ π β (π΄πΌπ))))) |
21 | 1, 2, 18, 20 | mpbir3and 1340 | 1 β’ (π β π(πΎβπ΄)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ wo 846 = wceq 1534 β wcel 2099 β wne 2937 class class class wbr 5148 βcfv 6548 (class class class)co 7420 Basecbs 17180 distcds 17242 TarskiGcstrkg 28244 Itvcitv 28250 LineGclng 28251 hlGchlg 28417 pInvGcmir 28469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9925 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-xnn0 12576 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14323 df-word 14498 df-concat 14554 df-s1 14579 df-s2 14832 df-s3 14833 df-trkgc 28265 df-trkgb 28266 df-trkgcb 28267 df-trkg 28270 df-cgrg 28328 df-hlg 28418 df-mir 28470 |
This theorem is referenced by: colhp 28587 sacgr 28648 |
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