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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 28416 | . . 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 28422 | . . 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 28243 | . . 3 β’ (π β π΄ β ((πβπ)πΌπ)) |
17 | 3, 6, 4, 7, 8, 5, 9, 10, 11 | mirbtwn 28413 | . . 3 β’ (π β π΄ β ((πβπ)πΌπ)) |
18 | 3, 4, 5, 12, 9, 13, 11, 14, 16, 17 | tgbtwnconn2 28331 | . 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 28357 | . 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 1339 | 1 β’ (π β π(πΎβπ΄)π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ wo 844 = wceq 1533 β wcel 2098 β wne 2934 class class class wbr 5141 βcfv 6536 (class class class)co 7404 Basecbs 17151 distcds 17213 TarskiGcstrkg 28182 Itvcitv 28188 LineGclng 28189 hlGchlg 28355 pInvGcmir 28407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-er 8702 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-xnn0 12546 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-hash 14294 df-word 14469 df-concat 14525 df-s1 14550 df-s2 14803 df-s3 14804 df-trkgc 28203 df-trkgb 28204 df-trkgcb 28205 df-trkg 28208 df-cgrg 28266 df-hlg 28356 df-mir 28408 |
This theorem is referenced by: colhp 28525 sacgr 28586 |
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