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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 11205 | . . . 4 ⊢ i ∈ ℂ | |
2 | mulcl 11230 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
4 | cosval 16107 | . . 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 11498 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
7 | efcl 16066 | . . . . 5 ⊢ (-𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) |
9 | efcl 16066 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) | |
10 | ixi 11881 | . . . . . . . 8 ⊢ (i · i) = -1 | |
11 | 10 | oveq1i 7436 | . . . . . . 7 ⊢ ((i · i) · 𝐴) = (-1 · 𝐴) |
12 | mulass 11234 | . . . . . . . 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 11693 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) | |
15 | 11, 13, 14 | 3eqtr3a 2792 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (i · 𝐴)) = -𝐴) |
16 | 15 | fveq2d 6906 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (i · 𝐴))) = (exp‘-𝐴)) |
17 | 1, 1 | mulneg1i 11698 | . . . . . . . . 9 ⊢ (-i · i) = -(i · i) |
18 | 10 | negeqi 11491 | . . . . . . . . 9 ⊢ -(i · i) = --1 |
19 | negneg1e1 12368 | . . . . . . . . 9 ⊢ --1 = 1 | |
20 | 17, 18, 19 | 3eqtri 2760 | . . . . . . . 8 ⊢ (-i · i) = 1 |
21 | 20 | oveq1i 7436 | . . . . . . 7 ⊢ ((-i · i) · 𝐴) = (1 · 𝐴) |
22 | negicn 11499 | . . . . . . . 8 ⊢ -i ∈ ℂ | |
23 | mulass 11234 | . . . . . . . 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 | mullid 11251 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
26 | 21, 24, 25 | 3eqtr3a 2792 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-i · (i · 𝐴)) = 𝐴) |
27 | 26 | fveq2d 6906 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (i · 𝐴))) = (exp‘𝐴)) |
28 | 16, 27 | oveq12d 7444 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) = ((exp‘-𝐴) + (exp‘𝐴))) |
29 | 8, 9, 28 | comraddd 11466 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) = ((exp‘𝐴) + (exp‘-𝐴))) |
30 | 29 | oveq1d 7441 | . 2 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) / 2) = (((exp‘𝐴) + (exp‘-𝐴)) / 2)) |
31 | 5, 30 | eqtrd 2768 | 1 ⊢ (𝐴 ∈ ℂ → (cos‘(i · 𝐴)) = (((exp‘𝐴) + (exp‘-𝐴)) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 1c1 11147 ici 11148 + caddc 11149 · cmul 11151 -cneg 11483 / cdiv 11909 2c2 12305 expce 16045 cosccos 16048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-ico 13370 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-fac 14273 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-cos 16054 |
This theorem is referenced by: rpcoshcl 16141 tanhlt1 16144 sinhpcosh 48249 |
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