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Mirrors > Home > MPE Home > Th. List > coshval | Structured version Visualization version GIF version |
Description: Value of the hyperbolic cosine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.) |
Ref | Expression |
---|---|
coshval | ⊢ (𝐴 ∈ ℂ → (cos‘(i · 𝐴)) = (((exp‘𝐴) + (exp‘-𝐴)) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 10667 | . . . 4 ⊢ i ∈ ℂ | |
2 | mulcl 10692 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
4 | cosval 15561 | . . 3 ⊢ ((i · 𝐴) ∈ ℂ → (cos‘(i · 𝐴)) = (((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) / 2)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘(i · 𝐴)) = (((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) / 2)) |
6 | negcl 10957 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
7 | efcl 15521 | . . . . 5 ⊢ (-𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) |
9 | efcl 15521 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) | |
10 | ixi 11340 | . . . . . . . 8 ⊢ (i · i) = -1 | |
11 | 10 | oveq1i 7174 | . . . . . . 7 ⊢ ((i · i) · 𝐴) = (-1 · 𝐴) |
12 | mulass 10696 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((i · i) · 𝐴) = (i · (i · 𝐴))) | |
13 | 1, 1, 12 | mp3an12 1452 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((i · i) · 𝐴) = (i · (i · 𝐴))) |
14 | mulm1 11152 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) | |
15 | 11, 13, 14 | 3eqtr3a 2797 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (i · 𝐴)) = -𝐴) |
16 | 15 | fveq2d 6672 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (i · 𝐴))) = (exp‘-𝐴)) |
17 | 1, 1 | mulneg1i 11157 | . . . . . . . . 9 ⊢ (-i · i) = -(i · i) |
18 | 10 | negeqi 10950 | . . . . . . . . 9 ⊢ -(i · i) = --1 |
19 | negneg1e1 11827 | . . . . . . . . 9 ⊢ --1 = 1 | |
20 | 17, 18, 19 | 3eqtri 2765 | . . . . . . . 8 ⊢ (-i · i) = 1 |
21 | 20 | oveq1i 7174 | . . . . . . 7 ⊢ ((-i · i) · 𝐴) = (1 · 𝐴) |
22 | negicn 10958 | . . . . . . . 8 ⊢ -i ∈ ℂ | |
23 | mulass 10696 | . . . . . . . 8 ⊢ ((-i ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((-i · i) · 𝐴) = (-i · (i · 𝐴))) | |
24 | 22, 1, 23 | mp3an12 1452 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((-i · i) · 𝐴) = (-i · (i · 𝐴))) |
25 | mulid2 10711 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
26 | 21, 24, 25 | 3eqtr3a 2797 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-i · (i · 𝐴)) = 𝐴) |
27 | 26 | fveq2d 6672 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (i · 𝐴))) = (exp‘𝐴)) |
28 | 16, 27 | oveq12d 7182 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) = ((exp‘-𝐴) + (exp‘𝐴))) |
29 | 8, 9, 28 | comraddd 10925 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) = ((exp‘𝐴) + (exp‘-𝐴))) |
30 | 29 | oveq1d 7179 | . 2 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) / 2) = (((exp‘𝐴) + (exp‘-𝐴)) / 2)) |
31 | 5, 30 | eqtrd 2773 | 1 ⊢ (𝐴 ∈ ℂ → (cos‘(i · 𝐴)) = (((exp‘𝐴) + (exp‘-𝐴)) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ‘cfv 6333 (class class class)co 7164 ℂcc 10606 1c1 10609 ici 10610 + caddc 10611 · cmul 10613 -cneg 10942 / cdiv 11368 2c2 11764 expce 15500 cosccos 15503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-pm 8433 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-sup 8972 df-inf 8973 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-n0 11970 df-z 12056 df-uz 12318 df-rp 12466 df-ico 12820 df-fz 12975 df-fzo 13118 df-fl 13246 df-seq 13454 df-exp 13515 df-fac 13719 df-hash 13776 df-shft 14509 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-limsup 14911 df-clim 14928 df-rlim 14929 df-sum 15129 df-ef 15506 df-cos 15509 |
This theorem is referenced by: rpcoshcl 15595 tanhlt1 15598 sinhpcosh 45879 |
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