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Mirrors > Home > MPE Home > Th. List > perprag | Structured version Visualization version GIF version |
Description: Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 10-Nov-2019.) |
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) |
perprag.1 | β’ (π β π΄ β π) |
perprag.2 | β’ (π β π΅ β π) |
perprag.3 | β’ (π β πΆ β (π΄πΏπ΅)) |
perprag.4 | β’ (π β π· β π) |
perprag.5 | β’ (π β (π΄πΏπ΅)(βGβπΊ)(πΆπΏπ·)) |
Ref | Expression |
---|---|
perprag | β’ (π β β¨βπ΄πΆπ·ββ© β (βGβπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2727 | . . . 4 β’ ((π β§ πΆ = π·) β π΄ = π΄) | |
2 | simpr 484 | . . . 4 β’ ((π β§ πΆ = π·) β πΆ = π·) | |
3 | eqidd 2727 | . . . 4 β’ ((π β§ πΆ = π·) β π· = π·) | |
4 | 1, 2, 3 | s3eqd 14819 | . . 3 β’ ((π β§ πΆ = π·) β β¨βπ΄πΆπ·ββ© = β¨βπ΄π·π·ββ©) |
5 | colperpex.p | . . . . 5 β’ π = (BaseβπΊ) | |
6 | colperpex.d | . . . . 5 β’ β = (distβπΊ) | |
7 | colperpex.i | . . . . 5 β’ πΌ = (ItvβπΊ) | |
8 | colperpex.l | . . . . 5 β’ πΏ = (LineGβπΊ) | |
9 | eqid 2726 | . . . . 5 β’ (pInvGβπΊ) = (pInvGβπΊ) | |
10 | colperpex.g | . . . . 5 β’ (π β πΊ β TarskiG) | |
11 | perprag.1 | . . . . 5 β’ (π β π΄ β π) | |
12 | perprag.4 | . . . . 5 β’ (π β π· β π) | |
13 | 5, 6, 7, 8, 9, 10, 11, 12, 12 | ragtrivb 28457 | . . . 4 β’ (π β β¨βπ΄π·π·ββ© β (βGβπΊ)) |
14 | 13 | adantr 480 | . . 3 β’ ((π β§ πΆ = π·) β β¨βπ΄π·π·ββ© β (βGβπΊ)) |
15 | 4, 14 | eqeltrd 2827 | . 2 β’ ((π β§ πΆ = π·) β β¨βπ΄πΆπ·ββ© β (βGβπΊ)) |
16 | 10 | adantr 480 | . . 3 β’ ((π β§ πΆ β π·) β πΊ β TarskiG) |
17 | perprag.2 | . . . . 5 β’ (π β π΅ β π) | |
18 | perprag.3 | . . . . . 6 β’ (π β πΆ β (π΄πΏπ΅)) | |
19 | 5, 8, 7, 10, 11, 17, 18 | tglngne 28305 | . . . . 5 β’ (π β π΄ β π΅) |
20 | 5, 7, 8, 10, 11, 17, 19 | tgelrnln 28385 | . . . 4 β’ (π β (π΄πΏπ΅) β ran πΏ) |
21 | 20 | adantr 480 | . . 3 β’ ((π β§ πΆ β π·) β (π΄πΏπ΅) β ran πΏ) |
22 | 5, 8, 7, 10, 20, 18 | tglnpt 28304 | . . . . 5 β’ (π β πΆ β π) |
23 | 22 | adantr 480 | . . . 4 β’ ((π β§ πΆ β π·) β πΆ β π) |
24 | 12 | adantr 480 | . . . 4 β’ ((π β§ πΆ β π·) β π· β π) |
25 | simpr 484 | . . . 4 β’ ((π β§ πΆ β π·) β πΆ β π·) | |
26 | 5, 7, 8, 16, 23, 24, 25 | tgelrnln 28385 | . . 3 β’ ((π β§ πΆ β π·) β (πΆπΏπ·) β ran πΏ) |
27 | 18 | adantr 480 | . . . 4 β’ ((π β§ πΆ β π·) β πΆ β (π΄πΏπ΅)) |
28 | 5, 7, 8, 16, 23, 24, 25 | tglinerflx1 28388 | . . . 4 β’ ((π β§ πΆ β π·) β πΆ β (πΆπΏπ·)) |
29 | 27, 28 | elind 4189 | . . 3 β’ ((π β§ πΆ β π·) β πΆ β ((π΄πΏπ΅) β© (πΆπΏπ·))) |
30 | 5, 7, 8, 10, 11, 17, 19 | tglinerflx1 28388 | . . . 4 β’ (π β π΄ β (π΄πΏπ΅)) |
31 | 30 | adantr 480 | . . 3 β’ ((π β§ πΆ β π·) β π΄ β (π΄πΏπ΅)) |
32 | 5, 7, 8, 16, 23, 24, 25 | tglinerflx2 28389 | . . 3 β’ ((π β§ πΆ β π·) β π· β (πΆπΏπ·)) |
33 | perprag.5 | . . . 4 β’ (π β (π΄πΏπ΅)(βGβπΊ)(πΆπΏπ·)) | |
34 | 33 | adantr 480 | . . 3 β’ ((π β§ πΆ β π·) β (π΄πΏπ΅)(βGβπΊ)(πΆπΏπ·)) |
35 | 5, 6, 7, 8, 16, 21, 26, 29, 31, 32, 34 | isperp2d 28471 | . 2 β’ ((π β§ πΆ β π·) β β¨βπ΄πΆπ·ββ© β (βGβπΊ)) |
36 | 15, 35 | pm2.61dane 3023 | 1 β’ (π β β¨βπ΄πΆπ·ββ© β (βGβπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2934 class class class wbr 5141 ran crn 5670 βcfv 6536 (class class class)co 7404 β¨βcs3 14797 Basecbs 17151 distcds 17213 TarskiGcstrkg 28182 Itvcitv 28188 LineGclng 28189 pInvGcmir 28407 βGcrag 28448 βGcperpg 28450 |
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-map 8821 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-mir 28408 df-rag 28449 df-perpg 28451 |
This theorem is referenced by: perpdragALT 28482 perpdrag 28483 colperpexlem3 28487 mideulem2 28489 opphllem 28490 opphllem5 28506 opphllem6 28507 trgcopy 28559 |
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