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Mirrors > Home > MPE Home > Th. List > cosargd | Structured version Visualization version GIF version |
Description: The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 25184. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
cosargd.1 | ⊢ (𝜑 → 𝑋 ∈ ℂ) |
cosargd.2 | ⊢ (𝜑 → 𝑋 ≠ 0) |
Ref | Expression |
---|---|
cosargd | ⊢ (𝜑 → (cos‘(ℑ‘(log‘𝑋))) = ((ℜ‘𝑋) / (abs‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cosargd.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℂ) | |
2 | 1 | cjcld 14549 | . . . 4 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ) |
3 | 1, 2 | addcld 10654 | . . 3 ⊢ (𝜑 → (𝑋 + (∗‘𝑋)) ∈ ℂ) |
4 | 1 | abscld 14790 | . . . 4 ⊢ (𝜑 → (abs‘𝑋) ∈ ℝ) |
5 | 4 | recnd 10663 | . . 3 ⊢ (𝜑 → (abs‘𝑋) ∈ ℂ) |
6 | 2cnd 11709 | . . 3 ⊢ (𝜑 → 2 ∈ ℂ) | |
7 | cosargd.2 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0) | |
8 | 1, 7 | absne0d 14801 | . . 3 ⊢ (𝜑 → (abs‘𝑋) ≠ 0) |
9 | 2ne0 11735 | . . . 4 ⊢ 2 ≠ 0 | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 2 ≠ 0) |
11 | 3, 5, 6, 8, 10 | divdiv32d 11435 | . 2 ⊢ (𝜑 → (((𝑋 + (∗‘𝑋)) / (abs‘𝑋)) / 2) = (((𝑋 + (∗‘𝑋)) / 2) / (abs‘𝑋))) |
12 | 1, 7 | logcld 25148 | . . . . . 6 ⊢ (𝜑 → (log‘𝑋) ∈ ℂ) |
13 | 12 | imcld 14548 | . . . . 5 ⊢ (𝜑 → (ℑ‘(log‘𝑋)) ∈ ℝ) |
14 | 13 | recnd 10663 | . . . 4 ⊢ (𝜑 → (ℑ‘(log‘𝑋)) ∈ ℂ) |
15 | cosval 15470 | . . . 4 ⊢ ((ℑ‘(log‘𝑋)) ∈ ℂ → (cos‘(ℑ‘(log‘𝑋))) = (((exp‘(i · (ℑ‘(log‘𝑋)))) + (exp‘(-i · (ℑ‘(log‘𝑋))))) / 2)) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → (cos‘(ℑ‘(log‘𝑋))) = (((exp‘(i · (ℑ‘(log‘𝑋)))) + (exp‘(-i · (ℑ‘(log‘𝑋))))) / 2)) |
17 | efiarg 25184 | . . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (exp‘(i · (ℑ‘(log‘𝑋)))) = (𝑋 / (abs‘𝑋))) | |
18 | 1, 7, 17 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (exp‘(i · (ℑ‘(log‘𝑋)))) = (𝑋 / (abs‘𝑋))) |
19 | ax-icn 10590 | . . . . . . . . . . 11 ⊢ i ∈ ℂ | |
20 | 19 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → i ∈ ℂ) |
21 | 20, 14 | mulcld 10655 | . . . . . . . . 9 ⊢ (𝜑 → (i · (ℑ‘(log‘𝑋))) ∈ ℂ) |
22 | efcj 15439 | . . . . . . . . 9 ⊢ ((i · (ℑ‘(log‘𝑋))) ∈ ℂ → (exp‘(∗‘(i · (ℑ‘(log‘𝑋))))) = (∗‘(exp‘(i · (ℑ‘(log‘𝑋)))))) | |
23 | 21, 22 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (exp‘(∗‘(i · (ℑ‘(log‘𝑋))))) = (∗‘(exp‘(i · (ℑ‘(log‘𝑋)))))) |
24 | 20, 14 | cjmuld 14574 | . . . . . . . . . 10 ⊢ (𝜑 → (∗‘(i · (ℑ‘(log‘𝑋)))) = ((∗‘i) · (∗‘(ℑ‘(log‘𝑋))))) |
25 | cji 14512 | . . . . . . . . . . . 12 ⊢ (∗‘i) = -i | |
26 | 25 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝜑 → (∗‘i) = -i) |
27 | 13 | cjred 14579 | . . . . . . . . . . 11 ⊢ (𝜑 → (∗‘(ℑ‘(log‘𝑋))) = (ℑ‘(log‘𝑋))) |
28 | 26, 27 | oveq12d 7168 | . . . . . . . . . 10 ⊢ (𝜑 → ((∗‘i) · (∗‘(ℑ‘(log‘𝑋)))) = (-i · (ℑ‘(log‘𝑋)))) |
29 | 24, 28 | eqtrd 2856 | . . . . . . . . 9 ⊢ (𝜑 → (∗‘(i · (ℑ‘(log‘𝑋)))) = (-i · (ℑ‘(log‘𝑋)))) |
30 | 29 | fveq2d 6668 | . . . . . . . 8 ⊢ (𝜑 → (exp‘(∗‘(i · (ℑ‘(log‘𝑋))))) = (exp‘(-i · (ℑ‘(log‘𝑋))))) |
31 | 18 | fveq2d 6668 | . . . . . . . 8 ⊢ (𝜑 → (∗‘(exp‘(i · (ℑ‘(log‘𝑋))))) = (∗‘(𝑋 / (abs‘𝑋)))) |
32 | 23, 30, 31 | 3eqtr3d 2864 | . . . . . . 7 ⊢ (𝜑 → (exp‘(-i · (ℑ‘(log‘𝑋)))) = (∗‘(𝑋 / (abs‘𝑋)))) |
33 | 1, 5, 8 | cjdivd 14576 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝑋 / (abs‘𝑋))) = ((∗‘𝑋) / (∗‘(abs‘𝑋)))) |
34 | 4 | cjred 14579 | . . . . . . . 8 ⊢ (𝜑 → (∗‘(abs‘𝑋)) = (abs‘𝑋)) |
35 | 34 | oveq2d 7166 | . . . . . . 7 ⊢ (𝜑 → ((∗‘𝑋) / (∗‘(abs‘𝑋))) = ((∗‘𝑋) / (abs‘𝑋))) |
36 | 32, 33, 35 | 3eqtrd 2860 | . . . . . 6 ⊢ (𝜑 → (exp‘(-i · (ℑ‘(log‘𝑋)))) = ((∗‘𝑋) / (abs‘𝑋))) |
37 | 18, 36 | oveq12d 7168 | . . . . 5 ⊢ (𝜑 → ((exp‘(i · (ℑ‘(log‘𝑋)))) + (exp‘(-i · (ℑ‘(log‘𝑋))))) = ((𝑋 / (abs‘𝑋)) + ((∗‘𝑋) / (abs‘𝑋)))) |
38 | 1, 2, 5, 8 | divdird 11448 | . . . . 5 ⊢ (𝜑 → ((𝑋 + (∗‘𝑋)) / (abs‘𝑋)) = ((𝑋 / (abs‘𝑋)) + ((∗‘𝑋) / (abs‘𝑋)))) |
39 | 37, 38 | eqtr4d 2859 | . . . 4 ⊢ (𝜑 → ((exp‘(i · (ℑ‘(log‘𝑋)))) + (exp‘(-i · (ℑ‘(log‘𝑋))))) = ((𝑋 + (∗‘𝑋)) / (abs‘𝑋))) |
40 | 39 | oveq1d 7165 | . . 3 ⊢ (𝜑 → (((exp‘(i · (ℑ‘(log‘𝑋)))) + (exp‘(-i · (ℑ‘(log‘𝑋))))) / 2) = (((𝑋 + (∗‘𝑋)) / (abs‘𝑋)) / 2)) |
41 | 16, 40 | eqtrd 2856 | . 2 ⊢ (𝜑 → (cos‘(ℑ‘(log‘𝑋))) = (((𝑋 + (∗‘𝑋)) / (abs‘𝑋)) / 2)) |
42 | reval 14459 | . . . 4 ⊢ (𝑋 ∈ ℂ → (ℜ‘𝑋) = ((𝑋 + (∗‘𝑋)) / 2)) | |
43 | 1, 42 | syl 17 | . . 3 ⊢ (𝜑 → (ℜ‘𝑋) = ((𝑋 + (∗‘𝑋)) / 2)) |
44 | 43 | oveq1d 7165 | . 2 ⊢ (𝜑 → ((ℜ‘𝑋) / (abs‘𝑋)) = (((𝑋 + (∗‘𝑋)) / 2) / (abs‘𝑋))) |
45 | 11, 41, 44 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → (cos‘(ℑ‘(log‘𝑋))) = ((ℜ‘𝑋) / (abs‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 0cc0 10531 ici 10533 + caddc 10534 · cmul 10536 -cneg 10865 / cdiv 11291 2c2 11686 ∗ccj 14449 ℜcre 14450 ℑcim 14451 abscabs 14587 expce 15409 cosccos 15412 logclog 25132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 |
This theorem is referenced by: cosarg0d 25186 cosangneg2d 25379 |
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