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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 24157. (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 13784 | . . . 4 ⊢ (𝜑 → (∗‘𝑋) ∈ ℂ) |
3 | 1, 2 | addcld 9938 | . . 3 ⊢ (𝜑 → (𝑋 + (∗‘𝑋)) ∈ ℂ) |
4 | 1 | abscld 14023 | . . . 4 ⊢ (𝜑 → (abs‘𝑋) ∈ ℝ) |
5 | 4 | recnd 9947 | . . 3 ⊢ (𝜑 → (abs‘𝑋) ∈ ℂ) |
6 | 2cnd 10970 | . . 3 ⊢ (𝜑 → 2 ∈ ℂ) | |
7 | cosargd.2 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0) | |
8 | 1, 7 | absne0d 14034 | . . 3 ⊢ (𝜑 → (abs‘𝑋) ≠ 0) |
9 | 2ne0 10990 | . . . 4 ⊢ 2 ≠ 0 | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 2 ≠ 0) |
11 | 3, 5, 6, 8, 10 | divdiv32d 10705 | . 2 ⊢ (𝜑 → (((𝑋 + (∗‘𝑋)) / (abs‘𝑋)) / 2) = (((𝑋 + (∗‘𝑋)) / 2) / (abs‘𝑋))) |
12 | 1, 7 | logcld 24121 | . . . . . 6 ⊢ (𝜑 → (log‘𝑋) ∈ ℂ) |
13 | 12 | imcld 13783 | . . . . 5 ⊢ (𝜑 → (ℑ‘(log‘𝑋)) ∈ ℝ) |
14 | 13 | recnd 9947 | . . . 4 ⊢ (𝜑 → (ℑ‘(log‘𝑋)) ∈ ℂ) |
15 | cosval 14692 | . . . 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 24157 | . . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (exp‘(i · (ℑ‘(log‘𝑋)))) = (𝑋 / (abs‘𝑋))) | |
18 | 1, 7, 17 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → (exp‘(i · (ℑ‘(log‘𝑋)))) = (𝑋 / (abs‘𝑋))) |
19 | ax-icn 9874 | . . . . . . . . . . 11 ⊢ i ∈ ℂ | |
20 | 19 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → i ∈ ℂ) |
21 | 20, 14 | mulcld 9939 | . . . . . . . . 9 ⊢ (𝜑 → (i · (ℑ‘(log‘𝑋))) ∈ ℂ) |
22 | efcj 14661 | . . . . . . . . 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 13809 | . . . . . . . . . 10 ⊢ (𝜑 → (∗‘(i · (ℑ‘(log‘𝑋)))) = ((∗‘i) · (∗‘(ℑ‘(log‘𝑋))))) |
25 | cji 13747 | . . . . . . . . . . . 12 ⊢ (∗‘i) = -i | |
26 | 25 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝜑 → (∗‘i) = -i) |
27 | 13 | cjred 13814 | . . . . . . . . . . 11 ⊢ (𝜑 → (∗‘(ℑ‘(log‘𝑋))) = (ℑ‘(log‘𝑋))) |
28 | 26, 27 | oveq12d 6567 | . . . . . . . . . 10 ⊢ (𝜑 → ((∗‘i) · (∗‘(ℑ‘(log‘𝑋)))) = (-i · (ℑ‘(log‘𝑋)))) |
29 | 24, 28 | eqtrd 2644 | . . . . . . . . 9 ⊢ (𝜑 → (∗‘(i · (ℑ‘(log‘𝑋)))) = (-i · (ℑ‘(log‘𝑋)))) |
30 | 29 | fveq2d 6107 | . . . . . . . 8 ⊢ (𝜑 → (exp‘(∗‘(i · (ℑ‘(log‘𝑋))))) = (exp‘(-i · (ℑ‘(log‘𝑋))))) |
31 | 18 | fveq2d 6107 | . . . . . . . 8 ⊢ (𝜑 → (∗‘(exp‘(i · (ℑ‘(log‘𝑋))))) = (∗‘(𝑋 / (abs‘𝑋)))) |
32 | 23, 30, 31 | 3eqtr3d 2652 | . . . . . . 7 ⊢ (𝜑 → (exp‘(-i · (ℑ‘(log‘𝑋)))) = (∗‘(𝑋 / (abs‘𝑋)))) |
33 | 1, 5, 8 | cjdivd 13811 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝑋 / (abs‘𝑋))) = ((∗‘𝑋) / (∗‘(abs‘𝑋)))) |
34 | 4 | cjred 13814 | . . . . . . . 8 ⊢ (𝜑 → (∗‘(abs‘𝑋)) = (abs‘𝑋)) |
35 | 34 | oveq2d 6565 | . . . . . . 7 ⊢ (𝜑 → ((∗‘𝑋) / (∗‘(abs‘𝑋))) = ((∗‘𝑋) / (abs‘𝑋))) |
36 | 32, 33, 35 | 3eqtrd 2648 | . . . . . 6 ⊢ (𝜑 → (exp‘(-i · (ℑ‘(log‘𝑋)))) = ((∗‘𝑋) / (abs‘𝑋))) |
37 | 18, 36 | oveq12d 6567 | . . . . 5 ⊢ (𝜑 → ((exp‘(i · (ℑ‘(log‘𝑋)))) + (exp‘(-i · (ℑ‘(log‘𝑋))))) = ((𝑋 / (abs‘𝑋)) + ((∗‘𝑋) / (abs‘𝑋)))) |
38 | 1, 2, 5, 8 | divdird 10718 | . . . . 5 ⊢ (𝜑 → ((𝑋 + (∗‘𝑋)) / (abs‘𝑋)) = ((𝑋 / (abs‘𝑋)) + ((∗‘𝑋) / (abs‘𝑋)))) |
39 | 37, 38 | eqtr4d 2647 | . . . 4 ⊢ (𝜑 → ((exp‘(i · (ℑ‘(log‘𝑋)))) + (exp‘(-i · (ℑ‘(log‘𝑋))))) = ((𝑋 + (∗‘𝑋)) / (abs‘𝑋))) |
40 | 39 | oveq1d 6564 | . . 3 ⊢ (𝜑 → (((exp‘(i · (ℑ‘(log‘𝑋)))) + (exp‘(-i · (ℑ‘(log‘𝑋))))) / 2) = (((𝑋 + (∗‘𝑋)) / (abs‘𝑋)) / 2)) |
41 | 16, 40 | eqtrd 2644 | . 2 ⊢ (𝜑 → (cos‘(ℑ‘(log‘𝑋))) = (((𝑋 + (∗‘𝑋)) / (abs‘𝑋)) / 2)) |
42 | reval 13694 | . . . 4 ⊢ (𝑋 ∈ ℂ → (ℜ‘𝑋) = ((𝑋 + (∗‘𝑋)) / 2)) | |
43 | 1, 42 | syl 17 | . . 3 ⊢ (𝜑 → (ℜ‘𝑋) = ((𝑋 + (∗‘𝑋)) / 2)) |
44 | 43 | oveq1d 6564 | . 2 ⊢ (𝜑 → ((ℜ‘𝑋) / (abs‘𝑋)) = (((𝑋 + (∗‘𝑋)) / 2) / (abs‘𝑋))) |
45 | 11, 41, 44 | 3eqtr4d 2654 | 1 ⊢ (𝜑 → (cos‘(ℑ‘(log‘𝑋))) = ((ℜ‘𝑋) / (abs‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 ici 9817 + caddc 9818 · cmul 9820 -cneg 10146 / cdiv 10563 2c2 10947 ∗ccj 13684 ℜcre 13685 ℑcim 13686 abscabs 13822 expce 14631 cosccos 14634 logclog 24105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 |
This theorem is referenced by: cosarg0d 24159 cosangneg2d 24337 |
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