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Mirrors > Home > MPE Home > Th. List > oacgr | Structured version Visualization version GIF version |
Description: Vertical angle theorem. Vertical, or opposite angles are the facing pair of angles formed when two lines intersect. Eudemus of Rhodes attributed the proof to Thales of Miletus. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. We follow the same path. Theorem 11.14 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 27-Sep-2020.) |
Ref | Expression |
---|---|
dfcgra2.p | β’ π = (BaseβπΊ) |
dfcgra2.i | β’ πΌ = (ItvβπΊ) |
dfcgra2.m | β’ β = (distβπΊ) |
dfcgra2.g | β’ (π β πΊ β TarskiG) |
dfcgra2.a | β’ (π β π΄ β π) |
dfcgra2.b | β’ (π β π΅ β π) |
dfcgra2.c | β’ (π β πΆ β π) |
dfcgra2.d | β’ (π β π· β π) |
dfcgra2.e | β’ (π β πΈ β π) |
dfcgra2.f | β’ (π β πΉ β π) |
oacgr.1 | β’ (π β π΅ β (π΄πΌπ·)) |
oacgr.2 | β’ (π β π΅ β (πΆπΌπΉ)) |
oacgr.3 | β’ (π β π΅ β π΄) |
oacgr.4 | β’ (π β π΅ β πΆ) |
oacgr.5 | β’ (π β π΅ β π·) |
oacgr.6 | β’ (π β π΅ β πΉ) |
Ref | Expression |
---|---|
oacgr | β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·π΅πΉββ©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcgra2.p | . 2 β’ π = (BaseβπΊ) | |
2 | dfcgra2.i | . 2 β’ πΌ = (ItvβπΊ) | |
3 | dfcgra2.g | . 2 β’ (π β πΊ β TarskiG) | |
4 | eqid 2728 | . 2 β’ (hlGβπΊ) = (hlGβπΊ) | |
5 | dfcgra2.a | . 2 β’ (π β π΄ β π) | |
6 | dfcgra2.b | . 2 β’ (π β π΅ β π) | |
7 | dfcgra2.c | . 2 β’ (π β πΆ β π) | |
8 | oacgr.3 | . . . 4 β’ (π β π΅ β π΄) | |
9 | 8 | necomd 2993 | . . 3 β’ (π β π΄ β π΅) |
10 | oacgr.4 | . . 3 β’ (π β π΅ β πΆ) | |
11 | 1, 2, 3, 4, 5, 6, 7, 9, 10 | cgraswap 28623 | . 2 β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπΆπ΅π΄ββ©) |
12 | dfcgra2.d | . 2 β’ (π β π· β π) | |
13 | dfcgra2.f | . 2 β’ (π β πΉ β π) | |
14 | dfcgra2.m | . . 3 β’ β = (distβπΊ) | |
15 | oacgr.6 | . . . . 5 β’ (π β π΅ β πΉ) | |
16 | 15 | necomd 2993 | . . . 4 β’ (π β πΉ β π΅) |
17 | 1, 2, 3, 4, 13, 6, 5, 16, 8 | cgraswap 28623 | . . 3 β’ (π β β¨βπΉπ΅π΄ββ©(cgrAβπΊ)β¨βπ΄π΅πΉββ©) |
18 | oacgr.2 | . . . 4 β’ (π β π΅ β (πΆπΌπΉ)) | |
19 | 1, 14, 2, 3, 7, 6, 13, 18 | tgbtwncom 28291 | . . 3 β’ (π β π΅ β (πΉπΌπΆ)) |
20 | oacgr.1 | . . 3 β’ (π β π΅ β (π΄πΌπ·)) | |
21 | oacgr.5 | . . 3 β’ (π β π΅ β π·) | |
22 | 1, 2, 14, 3, 13, 6, 5, 5, 6, 13, 7, 12, 17, 19, 20, 10, 21 | sacgr 28634 | . 2 β’ (π β β¨βπΆπ΅π΄ββ©(cgrAβπΊ)β¨βπ·π΅πΉββ©) |
23 | 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 11, 12, 6, 13, 22 | cgratr 28626 | 1 β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·π΅πΉββ©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β wne 2937 class class class wbr 5148 βcfv 6548 (class class class)co 7420 β¨βcs3 14825 Basecbs 17179 distcds 17241 TarskiGcstrkg 28230 Itvcitv 28236 hlGchlg 28403 cgrAccgra 28610 |
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 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-oadd 8490 df-er 8724 df-map 8846 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-s2 14831 df-s3 14832 df-trkgc 28251 df-trkgb 28252 df-trkgcb 28253 df-trkg 28256 df-cgrg 28314 df-leg 28386 df-hlg 28404 df-mir 28456 df-cgra 28611 |
This theorem is referenced by: (None) |
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