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Mirrors > Home > MPE Home > Th. List > cosangneg2d | Structured version Visualization version GIF version |
Description: The cosine of the angle between 𝑋 and -𝑌 is the negative of that between 𝑋 and 𝑌. If A, B and C are collinear points, this implies that the cosines of DBA and DBC sum to zero, i.e., that DBA and DBC are supplementary. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
ang.1 | ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥)))) |
cosangneg2d.1 | ⊢ (𝜑 → 𝑋 ∈ ℂ) |
cosangneg2d.2 | ⊢ (𝜑 → 𝑋 ≠ 0) |
cosangneg2d.3 | ⊢ (𝜑 → 𝑌 ∈ ℂ) |
cosangneg2d.4 | ⊢ (𝜑 → 𝑌 ≠ 0) |
Ref | Expression |
---|---|
cosangneg2d | ⊢ (𝜑 → (cos‘(𝑋𝐹-𝑌)) = -(cos‘(𝑋𝐹𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cosangneg2d.3 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℂ) | |
2 | cosangneg2d.1 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ) | |
3 | cosangneg2d.2 | . . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 0) | |
4 | 1, 2, 3 | divcld 11006 | . . . . 5 ⊢ (𝜑 → (𝑌 / 𝑋) ∈ ℂ) |
5 | 4 | recld 14141 | . . . 4 ⊢ (𝜑 → (ℜ‘(𝑌 / 𝑋)) ∈ ℝ) |
6 | 5 | recnd 10273 | . . 3 ⊢ (𝜑 → (ℜ‘(𝑌 / 𝑋)) ∈ ℂ) |
7 | 4 | abscld 14382 | . . . 4 ⊢ (𝜑 → (abs‘(𝑌 / 𝑋)) ∈ ℝ) |
8 | 7 | recnd 10273 | . . 3 ⊢ (𝜑 → (abs‘(𝑌 / 𝑋)) ∈ ℂ) |
9 | cosangneg2d.4 | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0) | |
10 | 1, 2, 9, 3 | divne0d 11022 | . . . 4 ⊢ (𝜑 → (𝑌 / 𝑋) ≠ 0) |
11 | 4, 10 | absne0d 14393 | . . 3 ⊢ (𝜑 → (abs‘(𝑌 / 𝑋)) ≠ 0) |
12 | 6, 8, 11 | divnegd 11019 | . 2 ⊢ (𝜑 → -((ℜ‘(𝑌 / 𝑋)) / (abs‘(𝑌 / 𝑋))) = (-(ℜ‘(𝑌 / 𝑋)) / (abs‘(𝑌 / 𝑋)))) |
13 | ang.1 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥)))) | |
14 | 13, 2, 3, 1, 9 | angvald 24754 | . . . . 5 ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) |
15 | 14 | fveq2d 6337 | . . . 4 ⊢ (𝜑 → (cos‘(𝑋𝐹𝑌)) = (cos‘(ℑ‘(log‘(𝑌 / 𝑋))))) |
16 | 4, 10 | cosargd 24574 | . . . 4 ⊢ (𝜑 → (cos‘(ℑ‘(log‘(𝑌 / 𝑋)))) = ((ℜ‘(𝑌 / 𝑋)) / (abs‘(𝑌 / 𝑋)))) |
17 | 15, 16 | eqtrd 2805 | . . 3 ⊢ (𝜑 → (cos‘(𝑋𝐹𝑌)) = ((ℜ‘(𝑌 / 𝑋)) / (abs‘(𝑌 / 𝑋)))) |
18 | 17 | negeqd 10480 | . 2 ⊢ (𝜑 → -(cos‘(𝑋𝐹𝑌)) = -((ℜ‘(𝑌 / 𝑋)) / (abs‘(𝑌 / 𝑋)))) |
19 | 1 | negcld 10584 | . . . . 5 ⊢ (𝜑 → -𝑌 ∈ ℂ) |
20 | 1, 9 | negne0d 10595 | . . . . 5 ⊢ (𝜑 → -𝑌 ≠ 0) |
21 | 13, 2, 3, 19, 20 | angvald 24754 | . . . 4 ⊢ (𝜑 → (𝑋𝐹-𝑌) = (ℑ‘(log‘(-𝑌 / 𝑋)))) |
22 | 21 | fveq2d 6337 | . . 3 ⊢ (𝜑 → (cos‘(𝑋𝐹-𝑌)) = (cos‘(ℑ‘(log‘(-𝑌 / 𝑋))))) |
23 | 19, 2, 3 | divcld 11006 | . . . 4 ⊢ (𝜑 → (-𝑌 / 𝑋) ∈ ℂ) |
24 | 19, 2, 20, 3 | divne0d 11022 | . . . 4 ⊢ (𝜑 → (-𝑌 / 𝑋) ≠ 0) |
25 | 23, 24 | cosargd 24574 | . . 3 ⊢ (𝜑 → (cos‘(ℑ‘(log‘(-𝑌 / 𝑋)))) = ((ℜ‘(-𝑌 / 𝑋)) / (abs‘(-𝑌 / 𝑋)))) |
26 | 1, 2, 3 | divnegd 11019 | . . . . . 6 ⊢ (𝜑 → -(𝑌 / 𝑋) = (-𝑌 / 𝑋)) |
27 | 26 | fveq2d 6337 | . . . . 5 ⊢ (𝜑 → (ℜ‘-(𝑌 / 𝑋)) = (ℜ‘(-𝑌 / 𝑋))) |
28 | 4 | renegd 14156 | . . . . 5 ⊢ (𝜑 → (ℜ‘-(𝑌 / 𝑋)) = -(ℜ‘(𝑌 / 𝑋))) |
29 | 27, 28 | eqtr3d 2807 | . . . 4 ⊢ (𝜑 → (ℜ‘(-𝑌 / 𝑋)) = -(ℜ‘(𝑌 / 𝑋))) |
30 | 26 | fveq2d 6337 | . . . . 5 ⊢ (𝜑 → (abs‘-(𝑌 / 𝑋)) = (abs‘(-𝑌 / 𝑋))) |
31 | 4 | absnegd 14395 | . . . . 5 ⊢ (𝜑 → (abs‘-(𝑌 / 𝑋)) = (abs‘(𝑌 / 𝑋))) |
32 | 30, 31 | eqtr3d 2807 | . . . 4 ⊢ (𝜑 → (abs‘(-𝑌 / 𝑋)) = (abs‘(𝑌 / 𝑋))) |
33 | 29, 32 | oveq12d 6813 | . . 3 ⊢ (𝜑 → ((ℜ‘(-𝑌 / 𝑋)) / (abs‘(-𝑌 / 𝑋))) = (-(ℜ‘(𝑌 / 𝑋)) / (abs‘(𝑌 / 𝑋)))) |
34 | 22, 25, 33 | 3eqtrd 2809 | . 2 ⊢ (𝜑 → (cos‘(𝑋𝐹-𝑌)) = (-(ℜ‘(𝑌 / 𝑋)) / (abs‘(𝑌 / 𝑋)))) |
35 | 12, 18, 34 | 3eqtr4rd 2816 | 1 ⊢ (𝜑 → (cos‘(𝑋𝐹-𝑌)) = -(cos‘(𝑋𝐹𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∖ cdif 3720 {csn 4317 ‘cfv 6030 (class class class)co 6795 ↦ cmpt2 6797 ℂcc 10139 0cc0 10141 -cneg 10472 / cdiv 10889 ℜcre 14044 ℑcim 14045 abscabs 14181 cosccos 15000 logclog 24521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7099 ax-inf2 8705 ax-cnex 10197 ax-resscn 10198 ax-1cn 10199 ax-icn 10200 ax-addcl 10201 ax-addrcl 10202 ax-mulcl 10203 ax-mulrcl 10204 ax-mulcom 10205 ax-addass 10206 ax-mulass 10207 ax-distr 10208 ax-i2m1 10209 ax-1ne0 10210 ax-1rid 10211 ax-rnegex 10212 ax-rrecex 10213 ax-cnre 10214 ax-pre-lttri 10215 ax-pre-lttrn 10216 ax-pre-ltadd 10217 ax-pre-mulgt0 10218 ax-pre-sup 10219 ax-addf 10220 ax-mulf 10221 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6756 df-ov 6798 df-oprab 6799 df-mpt2 6800 df-of 7047 df-om 7216 df-1st 7318 df-2nd 7319 df-supp 7450 df-wrecs 7562 df-recs 7624 df-rdg 7662 df-1o 7716 df-2o 7717 df-oadd 7720 df-er 7899 df-map 8014 df-pm 8015 df-ixp 8066 df-en 8113 df-dom 8114 df-sdom 8115 df-fin 8116 df-fsupp 8435 df-fi 8476 df-sup 8507 df-inf 8508 df-oi 8574 df-card 8968 df-cda 9195 df-pnf 10281 df-mnf 10282 df-xr 10283 df-ltxr 10284 df-le 10285 df-sub 10473 df-neg 10474 df-div 10890 df-nn 11226 df-2 11284 df-3 11285 df-4 11286 df-5 11287 df-6 11288 df-7 11289 df-8 11290 df-9 11291 df-n0 11499 df-z 11584 df-dec 11700 df-uz 11893 df-q 11996 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-ioo 12383 df-ioc 12384 df-ico 12385 df-icc 12386 df-fz 12533 df-fzo 12673 df-fl 12800 df-mod 12876 df-seq 13008 df-exp 13067 df-fac 13264 df-bc 13293 df-hash 13321 df-shft 14014 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-limsup 14409 df-clim 14426 df-rlim 14427 df-sum 14624 df-ef 15003 df-sin 15005 df-cos 15006 df-pi 15008 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-sca 16164 df-vsca 16165 df-ip 16166 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-hom 16173 df-cco 16174 df-rest 16290 df-topn 16291 df-0g 16309 df-gsum 16310 df-topgen 16311 df-pt 16312 df-prds 16315 df-xrs 16369 df-qtop 16374 df-imas 16375 df-xps 16377 df-mre 16453 df-mrc 16454 df-acs 16456 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-mulg 17748 df-cntz 17956 df-cmn 18401 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-fbas 19957 df-fg 19958 df-cnfld 19961 df-top 20918 df-topon 20935 df-topsp 20957 df-bases 20970 df-cld 21043 df-ntr 21044 df-cls 21045 df-nei 21122 df-lp 21160 df-perf 21161 df-cn 21251 df-cnp 21252 df-haus 21339 df-tx 21585 df-hmeo 21778 df-fil 21869 df-fm 21961 df-flim 21962 df-flf 21963 df-xms 22344 df-ms 22345 df-tms 22346 df-cncf 22900 df-limc 23849 df-dv 23850 df-log 24523 |
This theorem is referenced by: chordthmlem 24779 |
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