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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 11404 | . . . . 5 ⊢ (𝜑 → (𝑌 / 𝑋) ∈ ℂ) |
5 | 4 | recld 14541 | . . . 4 ⊢ (𝜑 → (ℜ‘(𝑌 / 𝑋)) ∈ ℝ) |
6 | 5 | recnd 10657 | . . 3 ⊢ (𝜑 → (ℜ‘(𝑌 / 𝑋)) ∈ ℂ) |
7 | 4 | abscld 14784 | . . . 4 ⊢ (𝜑 → (abs‘(𝑌 / 𝑋)) ∈ ℝ) |
8 | 7 | recnd 10657 | . . 3 ⊢ (𝜑 → (abs‘(𝑌 / 𝑋)) ∈ ℂ) |
9 | cosangneg2d.4 | . . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0) | |
10 | 1, 2, 9, 3 | divne0d 11420 | . . . 4 ⊢ (𝜑 → (𝑌 / 𝑋) ≠ 0) |
11 | 4, 10 | absne0d 14795 | . . 3 ⊢ (𝜑 → (abs‘(𝑌 / 𝑋)) ≠ 0) |
12 | 6, 8, 11 | divnegd 11417 | . 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 25309 | . . . . 5 ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) |
15 | 14 | fveq2d 6667 | . . . 4 ⊢ (𝜑 → (cos‘(𝑋𝐹𝑌)) = (cos‘(ℑ‘(log‘(𝑌 / 𝑋))))) |
16 | 4, 10 | cosargd 25118 | . . . 4 ⊢ (𝜑 → (cos‘(ℑ‘(log‘(𝑌 / 𝑋)))) = ((ℜ‘(𝑌 / 𝑋)) / (abs‘(𝑌 / 𝑋)))) |
17 | 15, 16 | eqtrd 2853 | . . 3 ⊢ (𝜑 → (cos‘(𝑋𝐹𝑌)) = ((ℜ‘(𝑌 / 𝑋)) / (abs‘(𝑌 / 𝑋)))) |
18 | 17 | negeqd 10868 | . 2 ⊢ (𝜑 → -(cos‘(𝑋𝐹𝑌)) = -((ℜ‘(𝑌 / 𝑋)) / (abs‘(𝑌 / 𝑋)))) |
19 | 1 | negcld 10972 | . . . . 5 ⊢ (𝜑 → -𝑌 ∈ ℂ) |
20 | 1, 9 | negne0d 10983 | . . . . 5 ⊢ (𝜑 → -𝑌 ≠ 0) |
21 | 13, 2, 3, 19, 20 | angvald 25309 | . . . 4 ⊢ (𝜑 → (𝑋𝐹-𝑌) = (ℑ‘(log‘(-𝑌 / 𝑋)))) |
22 | 21 | fveq2d 6667 | . . 3 ⊢ (𝜑 → (cos‘(𝑋𝐹-𝑌)) = (cos‘(ℑ‘(log‘(-𝑌 / 𝑋))))) |
23 | 19, 2, 3 | divcld 11404 | . . . 4 ⊢ (𝜑 → (-𝑌 / 𝑋) ∈ ℂ) |
24 | 19, 2, 20, 3 | divne0d 11420 | . . . 4 ⊢ (𝜑 → (-𝑌 / 𝑋) ≠ 0) |
25 | 23, 24 | cosargd 25118 | . . 3 ⊢ (𝜑 → (cos‘(ℑ‘(log‘(-𝑌 / 𝑋)))) = ((ℜ‘(-𝑌 / 𝑋)) / (abs‘(-𝑌 / 𝑋)))) |
26 | 1, 2, 3 | divnegd 11417 | . . . . . 6 ⊢ (𝜑 → -(𝑌 / 𝑋) = (-𝑌 / 𝑋)) |
27 | 26 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → (ℜ‘-(𝑌 / 𝑋)) = (ℜ‘(-𝑌 / 𝑋))) |
28 | 4 | renegd 14556 | . . . . 5 ⊢ (𝜑 → (ℜ‘-(𝑌 / 𝑋)) = -(ℜ‘(𝑌 / 𝑋))) |
29 | 27, 28 | eqtr3d 2855 | . . . 4 ⊢ (𝜑 → (ℜ‘(-𝑌 / 𝑋)) = -(ℜ‘(𝑌 / 𝑋))) |
30 | 26 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → (abs‘-(𝑌 / 𝑋)) = (abs‘(-𝑌 / 𝑋))) |
31 | 4 | absnegd 14797 | . . . . 5 ⊢ (𝜑 → (abs‘-(𝑌 / 𝑋)) = (abs‘(𝑌 / 𝑋))) |
32 | 30, 31 | eqtr3d 2855 | . . . 4 ⊢ (𝜑 → (abs‘(-𝑌 / 𝑋)) = (abs‘(𝑌 / 𝑋))) |
33 | 29, 32 | oveq12d 7163 | . . 3 ⊢ (𝜑 → ((ℜ‘(-𝑌 / 𝑋)) / (abs‘(-𝑌 / 𝑋))) = (-(ℜ‘(𝑌 / 𝑋)) / (abs‘(𝑌 / 𝑋)))) |
34 | 22, 25, 33 | 3eqtrd 2857 | . 2 ⊢ (𝜑 → (cos‘(𝑋𝐹-𝑌)) = (-(ℜ‘(𝑌 / 𝑋)) / (abs‘(𝑌 / 𝑋)))) |
35 | 12, 18, 34 | 3eqtr4rd 2864 | 1 ⊢ (𝜑 → (cos‘(𝑋𝐹-𝑌)) = -(cos‘(𝑋𝐹𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∖ cdif 3930 {csn 4557 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ℂcc 10523 0cc0 10525 -cneg 10859 / cdiv 11285 ℜcre 14444 ℑcim 14445 abscabs 14581 cosccos 15406 logclog 25065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 |
This theorem is referenced by: chordthmlem 25337 |
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