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Mirrors > Home > MPE Home > Th. List > hlperpnel | Structured version Visualization version GIF version |
Description: A point on a half-line which is perpendicular to a line cannot be on that line. (Contributed by Thierry Arnoux, 1-Mar-2020.) |
Ref | Expression |
---|---|
colperpex.p | β’ π = (BaseβπΊ) |
colperpex.d | β’ β = (distβπΊ) |
colperpex.i | β’ πΌ = (ItvβπΊ) |
colperpex.l | β’ πΏ = (LineGβπΊ) |
colperpex.g | β’ (π β πΊ β TarskiG) |
hlperpnel.a | β’ (π β π΄ β ran πΏ) |
hlperpnel.k | β’ πΎ = (hlGβπΊ) |
hlperpnel.1 | β’ (π β π β π΄) |
hlperpnel.2 | β’ (π β π β π) |
hlperpnel.3 | β’ (π β π β π) |
hlperpnel.4 | β’ (π β π΄(βGβπΊ)(ππΏπ)) |
hlperpnel.5 | β’ (π β π(πΎβπ)π) |
Ref | Expression |
---|---|
hlperpnel | β’ (π β Β¬ π β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | colperpex.p | . 2 β’ π = (BaseβπΊ) | |
2 | colperpex.d | . 2 β’ β = (distβπΊ) | |
3 | colperpex.i | . 2 β’ πΌ = (ItvβπΊ) | |
4 | colperpex.l | . 2 β’ πΏ = (LineGβπΊ) | |
5 | colperpex.g | . 2 β’ (π β πΊ β TarskiG) | |
6 | hlperpnel.a | . 2 β’ (π β π΄ β ran πΏ) | |
7 | hlperpnel.1 | . 2 β’ (π β π β π΄) | |
8 | hlperpnel.3 | . 2 β’ (π β π β π) | |
9 | 1, 4, 3, 5, 6, 7 | tglnpt 27199 | . . . 4 β’ (π β π β π) |
10 | hlperpnel.2 | . . . 4 β’ (π β π β π) | |
11 | hlperpnel.4 | . . . . . 6 β’ (π β π΄(βGβπΊ)(ππΏπ)) | |
12 | 4, 5, 11 | perpln2 27361 | . . . . 5 β’ (π β (ππΏπ) β ran πΏ) |
13 | 1, 3, 4, 5, 9, 10, 12 | tglnne 27278 | . . . 4 β’ (π β π β π) |
14 | hlperpnel.k | . . . . 5 β’ πΎ = (hlGβπΊ) | |
15 | hlperpnel.5 | . . . . 5 β’ (π β π(πΎβπ)π) | |
16 | 1, 3, 14, 10, 8, 9, 5, 15 | hlne2 27256 | . . . 4 β’ (π β π β π) |
17 | 1, 3, 14, 10, 8, 9, 5, 4, 15 | hlln 27257 | . . . . 5 β’ (π β π β (ππΏπ)) |
18 | 1, 3, 4, 5, 9, 10, 8, 13, 17, 16 | lnrot1 27273 | . . . 4 β’ (π β π β (ππΏπ)) |
19 | 1, 3, 4, 5, 9, 10, 13, 8, 16, 18 | tglineelsb2 27282 | . . 3 β’ (π β (ππΏπ) = (ππΏπ)) |
20 | 1, 2, 3, 4, 5, 6, 12, 11 | perpcom 27363 | . . 3 β’ (π β (ππΏπ)(βGβπΊ)π΄) |
21 | 19, 20 | eqbrtrrd 5116 | . 2 β’ (π β (ππΏπ)(βGβπΊ)π΄) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 21 | footne 27373 | 1 β’ (π β Β¬ π β π΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 = wceq 1540 β wcel 2105 class class class wbr 5092 ran crn 5621 βcfv 6479 (class class class)co 7337 Basecbs 17009 distcds 17068 TarskiGcstrkg 27077 Itvcitv 27083 LineGclng 27084 hlGchlg 27250 βGcperpg 27345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-oadd 8371 df-er 8569 df-map 8688 df-pm 8689 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-dju 9758 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-n0 12335 df-xnn0 12407 df-z 12421 df-uz 12684 df-fz 13341 df-fzo 13484 df-hash 14146 df-word 14318 df-concat 14374 df-s1 14400 df-s2 14660 df-s3 14661 df-trkgc 27098 df-trkgb 27099 df-trkgcb 27100 df-trkg 27103 df-cgrg 27161 df-hlg 27251 df-mir 27303 df-rag 27344 df-perpg 27346 |
This theorem is referenced by: opphllem5 27401 |
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