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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 10590 | . . . 4 ⊢ i ∈ ℂ | |
2 | mulcl 10615 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
4 | cosval 15470 | . . 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 10880 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
7 | efcl 15430 | . . . . 5 ⊢ (-𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) |
9 | efcl 15430 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) | |
10 | ixi 11263 | . . . . . . . 8 ⊢ (i · i) = -1 | |
11 | 10 | oveq1i 7160 | . . . . . . 7 ⊢ ((i · i) · 𝐴) = (-1 · 𝐴) |
12 | mulass 10619 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((i · i) · 𝐴) = (i · (i · 𝐴))) | |
13 | 1, 1, 12 | mp3an12 1447 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((i · i) · 𝐴) = (i · (i · 𝐴))) |
14 | mulm1 11075 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) | |
15 | 11, 13, 14 | 3eqtr3a 2880 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (i · 𝐴)) = -𝐴) |
16 | 15 | fveq2d 6669 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (i · 𝐴))) = (exp‘-𝐴)) |
17 | 1, 1 | mulneg1i 11080 | . . . . . . . . 9 ⊢ (-i · i) = -(i · i) |
18 | 10 | negeqi 10873 | . . . . . . . . 9 ⊢ -(i · i) = --1 |
19 | negneg1e1 11749 | . . . . . . . . 9 ⊢ --1 = 1 | |
20 | 17, 18, 19 | 3eqtri 2848 | . . . . . . . 8 ⊢ (-i · i) = 1 |
21 | 20 | oveq1i 7160 | . . . . . . 7 ⊢ ((-i · i) · 𝐴) = (1 · 𝐴) |
22 | negicn 10881 | . . . . . . . 8 ⊢ -i ∈ ℂ | |
23 | mulass 10619 | . . . . . . . 8 ⊢ ((-i ∈ ℂ ∧ i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((-i · i) · 𝐴) = (-i · (i · 𝐴))) | |
24 | 22, 1, 23 | mp3an12 1447 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((-i · i) · 𝐴) = (-i · (i · 𝐴))) |
25 | mulid2 10634 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
26 | 21, 24, 25 | 3eqtr3a 2880 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-i · (i · 𝐴)) = 𝐴) |
27 | 26 | fveq2d 6669 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (i · 𝐴))) = (exp‘𝐴)) |
28 | 16, 27 | oveq12d 7168 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) = ((exp‘-𝐴) + (exp‘𝐴))) |
29 | 8, 9, 28 | comraddd 10848 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) = ((exp‘𝐴) + (exp‘-𝐴))) |
30 | 29 | oveq1d 7165 | . 2 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) / 2) = (((exp‘𝐴) + (exp‘-𝐴)) / 2)) |
31 | 5, 30 | eqtrd 2856 | 1 ⊢ (𝐴 ∈ ℂ → (cos‘(i · 𝐴)) = (((exp‘𝐴) + (exp‘-𝐴)) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 1c1 10532 ici 10533 + caddc 10534 · cmul 10536 -cneg 10865 / cdiv 11291 2c2 11686 expce 15409 cosccos 15412 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 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-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ico 12738 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-fac 13628 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-cos 15418 |
This theorem is referenced by: rpcoshcl 15504 tanhlt1 15507 sinhpcosh 44832 |
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