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Mirrors > Home > MPE Home > Th. List > footne | Structured version Visualization version GIF version |
Description: Uniqueness of the foot point. (Contributed by Thierry Arnoux, 28-Feb-2020.) |
Ref | Expression |
---|---|
isperp.p | β’ π = (BaseβπΊ) |
isperp.d | β’ β = (distβπΊ) |
isperp.i | β’ πΌ = (ItvβπΊ) |
isperp.l | β’ πΏ = (LineGβπΊ) |
isperp.g | β’ (π β πΊ β TarskiG) |
isperp.a | β’ (π β π΄ β ran πΏ) |
footne.x | β’ (π β π β π΄) |
footne.y | β’ (π β π β π) |
footne.1 | β’ (π β (ππΏπ)(βGβπΊ)π΄) |
Ref | Expression |
---|---|
footne | β’ (π β Β¬ π β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isperp.p | . . . 4 β’ π = (BaseβπΊ) | |
2 | isperp.i | . . . 4 β’ πΌ = (ItvβπΊ) | |
3 | isperp.l | . . . 4 β’ πΏ = (LineGβπΊ) | |
4 | isperp.g | . . . . 5 β’ (π β πΊ β TarskiG) | |
5 | 4 | adantr 480 | . . . 4 β’ ((π β§ π β π΄) β πΊ β TarskiG) |
6 | isperp.a | . . . . 5 β’ (π β π΄ β ran πΏ) | |
7 | 6 | adantr 480 | . . . 4 β’ ((π β§ π β π΄) β π΄ β ran πΏ) |
8 | footne.1 | . . . . . 6 β’ (π β (ππΏπ)(βGβπΊ)π΄) | |
9 | 3, 4, 8 | perpln1 28455 | . . . . 5 β’ (π β (ππΏπ) β ran πΏ) |
10 | 9 | adantr 480 | . . . 4 β’ ((π β§ π β π΄) β (ππΏπ) β ran πΏ) |
11 | isperp.d | . . . . . . 7 β’ β = (distβπΊ) | |
12 | 1, 11, 2, 3, 4, 9, 6, 8 | perpneq 28459 | . . . . . 6 β’ (π β (ππΏπ) β π΄) |
13 | 12 | necomd 2988 | . . . . 5 β’ (π β π΄ β (ππΏπ)) |
14 | 13 | adantr 480 | . . . 4 β’ ((π β§ π β π΄) β π΄ β (ππΏπ)) |
15 | footne.x | . . . . . 6 β’ (π β π β π΄) | |
16 | 15 | adantr 480 | . . . . 5 β’ ((π β§ π β π΄) β π β π΄) |
17 | 1, 3, 2, 4, 6, 15 | tglnpt 28294 | . . . . . . 7 β’ (π β π β π) |
18 | footne.y | . . . . . . 7 β’ (π β π β π) | |
19 | 1, 2, 3, 4, 17, 18, 9 | tglnne 28373 | . . . . . . 7 β’ (π β π β π) |
20 | 1, 2, 3, 4, 17, 18, 19 | tglinerflx1 28378 | . . . . . 6 β’ (π β π β (ππΏπ)) |
21 | 20 | adantr 480 | . . . . 5 β’ ((π β§ π β π΄) β π β (ππΏπ)) |
22 | 16, 21 | elind 4187 | . . . 4 β’ ((π β§ π β π΄) β π β (π΄ β© (ππΏπ))) |
23 | simpr 484 | . . . . 5 β’ ((π β§ π β π΄) β π β π΄) | |
24 | 1, 2, 3, 4, 17, 18, 19 | tglinerflx2 28379 | . . . . . 6 β’ (π β π β (ππΏπ)) |
25 | 24 | adantr 480 | . . . . 5 β’ ((π β§ π β π΄) β π β (ππΏπ)) |
26 | 23, 25 | elind 4187 | . . . 4 β’ ((π β§ π β π΄) β π β (π΄ β© (ππΏπ))) |
27 | 1, 2, 3, 5, 7, 10, 14, 22, 26 | tglineineq 28388 | . . 3 β’ ((π β§ π β π΄) β π = π) |
28 | 19 | adantr 480 | . . 3 β’ ((π β§ π β π΄) β π β π) |
29 | 27, 28 | pm2.21ddne 3018 | . 2 β’ ((π β§ π β π΄) β Β¬ π β π΄) |
30 | 29 | pm2.01da 796 | 1 β’ (π β Β¬ π β π΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2932 class class class wbr 5139 ran crn 5668 βcfv 6534 (class class class)co 7402 Basecbs 17149 distcds 17211 TarskiGcstrkg 28172 Itvcitv 28178 LineGclng 28179 βGcperpg 28440 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-dju 9893 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-xnn0 12544 df-z 12558 df-uz 12822 df-fz 13486 df-fzo 13629 df-hash 14292 df-word 14467 df-concat 14523 df-s1 14548 df-s2 14801 df-s3 14802 df-trkgc 28193 df-trkgb 28194 df-trkgcb 28195 df-trkg 28198 df-cgrg 28256 df-mir 28398 df-rag 28439 df-perpg 28441 |
This theorem is referenced by: footeq 28469 hlperpnel 28470 oppperpex 28498 |
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