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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 | ⊢ 𝑃 = (Base‘𝐺) |
dfcgra2.i | ⊢ 𝐼 = (Itv‘𝐺) |
dfcgra2.m | ⊢ − = (dist‘𝐺) |
dfcgra2.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
dfcgra2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
dfcgra2.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
dfcgra2.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
dfcgra2.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
dfcgra2.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
dfcgra2.f | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
oacgr.1 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) |
oacgr.2 | ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐹)) |
oacgr.3 | ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
oacgr.4 | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
oacgr.5 | ⊢ (𝜑 → 𝐵 ≠ 𝐷) |
oacgr.6 | ⊢ (𝜑 → 𝐵 ≠ 𝐹) |
Ref | Expression |
---|---|
oacgr | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐵𝐹”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcgra2.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | dfcgra2.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
3 | dfcgra2.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
4 | eqid 2777 | . 2 ⊢ (hlG‘𝐺) = (hlG‘𝐺) | |
5 | dfcgra2.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | dfcgra2.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | dfcgra2.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
8 | oacgr.3 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴) | |
9 | 8 | necomd 3023 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
10 | oacgr.4 | . . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
11 | 1, 2, 3, 4, 5, 6, 7, 9, 10 | cgraswap 26168 | . 2 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐶𝐵𝐴”〉) |
12 | dfcgra2.d | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
13 | dfcgra2.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
14 | dfcgra2.m | . . 3 ⊢ − = (dist‘𝐺) | |
15 | oacgr.6 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐹) | |
16 | 15 | necomd 3023 | . . . 4 ⊢ (𝜑 → 𝐹 ≠ 𝐵) |
17 | 1, 2, 3, 4, 13, 6, 5, 16, 8 | cgraswap 26168 | . . 3 ⊢ (𝜑 → 〈“𝐹𝐵𝐴”〉(cgrA‘𝐺)〈“𝐴𝐵𝐹”〉) |
18 | oacgr.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐹)) | |
19 | 1, 14, 2, 3, 7, 6, 13, 18 | tgbtwncom 25839 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐹𝐼𝐶)) |
20 | oacgr.1 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | |
21 | oacgr.5 | . . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐷) | |
22 | 1, 2, 14, 3, 13, 6, 5, 5, 6, 13, 7, 12, 17, 19, 20, 10, 21 | sacgr 26179 | . 2 ⊢ (𝜑 → 〈“𝐶𝐵𝐴”〉(cgrA‘𝐺)〈“𝐷𝐵𝐹”〉) |
23 | 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 11, 12, 6, 13, 22 | cgratr 26171 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐵𝐹”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 〈“cs3 13993 Basecbs 16255 distcds 16347 TarskiGcstrkg 25781 Itvcitv 25787 hlGchlg 25951 cgrAccgra 26155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-xnn0 11715 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-hash 13436 df-word 13600 df-concat 13661 df-s1 13686 df-s2 13999 df-s3 14000 df-trkgc 25799 df-trkgb 25800 df-trkgcb 25801 df-trkg 25804 df-cgrg 25862 df-leg 25934 df-hlg 25952 df-mir 26004 df-cgra 26156 |
This theorem is referenced by: (None) |
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