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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 10596 | . . . 4 ⊢ i ∈ ℂ | |
2 | mulcl 10621 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
4 | cosval 15476 | . . 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 10886 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
7 | efcl 15436 | . . . . 5 ⊢ (-𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) |
9 | efcl 15436 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) | |
10 | ixi 11269 | . . . . . . . 8 ⊢ (i · i) = -1 | |
11 | 10 | oveq1i 7166 | . . . . . . 7 ⊢ ((i · i) · 𝐴) = (-1 · 𝐴) |
12 | mulass 10625 | . . . . . . . 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 11081 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) | |
15 | 11, 13, 14 | 3eqtr3a 2880 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (i · 𝐴)) = -𝐴) |
16 | 15 | fveq2d 6674 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (i · 𝐴))) = (exp‘-𝐴)) |
17 | 1, 1 | mulneg1i 11086 | . . . . . . . . 9 ⊢ (-i · i) = -(i · i) |
18 | 10 | negeqi 10879 | . . . . . . . . 9 ⊢ -(i · i) = --1 |
19 | negneg1e1 11756 | . . . . . . . . 9 ⊢ --1 = 1 | |
20 | 17, 18, 19 | 3eqtri 2848 | . . . . . . . 8 ⊢ (-i · i) = 1 |
21 | 20 | oveq1i 7166 | . . . . . . 7 ⊢ ((-i · i) · 𝐴) = (1 · 𝐴) |
22 | negicn 10887 | . . . . . . . 8 ⊢ -i ∈ ℂ | |
23 | mulass 10625 | . . . . . . . 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 10640 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
26 | 21, 24, 25 | 3eqtr3a 2880 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-i · (i · 𝐴)) = 𝐴) |
27 | 26 | fveq2d 6674 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (i · 𝐴))) = (exp‘𝐴)) |
28 | 16, 27 | oveq12d 7174 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) = ((exp‘-𝐴) + (exp‘𝐴))) |
29 | 8, 9, 28 | comraddd 10854 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) = ((exp‘𝐴) + (exp‘-𝐴))) |
30 | 29 | oveq1d 7171 | . 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 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 1c1 10538 ici 10539 + caddc 10540 · cmul 10542 -cneg 10871 / cdiv 11297 2c2 11693 expce 15415 cosccos 15418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-ico 12745 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-fac 13635 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-cos 15424 |
This theorem is referenced by: rpcoshcl 15510 tanhlt1 15513 sinhpcosh 44859 |
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