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Mirrors > Home > ILE Home > Th. List > cosval | GIF version |
Description: Value of the cosine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.) |
Ref | Expression |
---|---|
cosval | ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 7937 | . . . . . . 7 ⊢ i ∈ ℂ | |
2 | 1 | a1i 9 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → i ∈ ℂ) |
3 | id 19 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
4 | 2, 3 | mulcld 8009 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
5 | efcl 11707 | . . . . 5 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ) |
7 | negicn 8189 | . . . . . . 7 ⊢ -i ∈ ℂ | |
8 | 7 | a1i 9 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → -i ∈ ℂ) |
9 | 8, 3 | mulcld 8009 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
10 | efcl 11707 | . . . . 5 ⊢ ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) | |
11 | 9, 10 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ) |
12 | 6, 11 | addcld 8008 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) ∈ ℂ) |
13 | 12 | halfcld 9194 | . 2 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2) ∈ ℂ) |
14 | oveq2 5905 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
15 | 14 | fveq2d 5538 | . . . . 5 ⊢ (𝑥 = 𝐴 → (exp‘(i · 𝑥)) = (exp‘(i · 𝐴))) |
16 | oveq2 5905 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (-i · 𝑥) = (-i · 𝐴)) | |
17 | 16 | fveq2d 5538 | . . . . 5 ⊢ (𝑥 = 𝐴 → (exp‘(-i · 𝑥)) = (exp‘(-i · 𝐴))) |
18 | 15, 17 | oveq12d 5915 | . . . 4 ⊢ (𝑥 = 𝐴 → ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) = ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴)))) |
19 | 18 | oveq1d 5912 | . . 3 ⊢ (𝑥 = 𝐴 → (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) |
20 | df-cos 11694 | . . 3 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) | |
21 | 19, 20 | fvmptg 5613 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2) ∈ ℂ) → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) |
22 | 13, 21 | mpdan 421 | 1 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 ‘cfv 5235 (class class class)co 5897 ℂcc 7840 ici 7844 + caddc 7845 · cmul 7847 -cneg 8160 / cdiv 8660 2c2 9001 expce 11685 cosccos 11688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 ax-arch 7961 ax-caucvg 7962 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-irdg 6396 df-frec 6417 df-1o 6442 df-oadd 6446 df-er 6560 df-en 6768 df-dom 6769 df-fin 6770 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-n0 9208 df-z 9285 df-uz 9560 df-q 9652 df-rp 9686 df-ico 9926 df-fz 10041 df-fzo 10175 df-seqfrec 10479 df-exp 10554 df-fac 10741 df-ihash 10791 df-cj 10886 df-re 10887 df-im 10888 df-rsqrt 11042 df-abs 11043 df-clim 11322 df-sumdc 11397 df-ef 11691 df-cos 11694 |
This theorem is referenced by: tanval2ap 11756 tanval3ap 11757 recosval 11759 cosneg 11770 efival 11775 cosadd 11780 cosper 14708 |
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