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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 11171 | . . . 4 ⊢ i ∈ ℂ | |
2 | mulcl 11196 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 687 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
4 | cosval 16073 | . . 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 11464 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
7 | efcl 16032 | . . . . 5 ⊢ (-𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) |
9 | efcl 16032 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) | |
10 | ixi 11847 | . . . . . . . 8 ⊢ (i · i) = -1 | |
11 | 10 | oveq1i 7415 | . . . . . . 7 ⊢ ((i · i) · 𝐴) = (-1 · 𝐴) |
12 | mulass 11200 | . . . . . . . 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 11659 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) | |
15 | 11, 13, 14 | 3eqtr3a 2790 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (i · 𝐴)) = -𝐴) |
16 | 15 | fveq2d 6889 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (i · 𝐴))) = (exp‘-𝐴)) |
17 | 1, 1 | mulneg1i 11664 | . . . . . . . . 9 ⊢ (-i · i) = -(i · i) |
18 | 10 | negeqi 11457 | . . . . . . . . 9 ⊢ -(i · i) = --1 |
19 | negneg1e1 12334 | . . . . . . . . 9 ⊢ --1 = 1 | |
20 | 17, 18, 19 | 3eqtri 2758 | . . . . . . . 8 ⊢ (-i · i) = 1 |
21 | 20 | oveq1i 7415 | . . . . . . 7 ⊢ ((-i · i) · 𝐴) = (1 · 𝐴) |
22 | negicn 11465 | . . . . . . . 8 ⊢ -i ∈ ℂ | |
23 | mulass 11200 | . . . . . . . 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 11217 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
26 | 21, 24, 25 | 3eqtr3a 2790 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-i · (i · 𝐴)) = 𝐴) |
27 | 26 | fveq2d 6889 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(-i · (i · 𝐴))) = (exp‘𝐴)) |
28 | 16, 27 | oveq12d 7423 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) = ((exp‘-𝐴) + (exp‘𝐴))) |
29 | 8, 9, 28 | comraddd 11432 | . . 3 ⊢ (𝐴 ∈ ℂ → ((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) = ((exp‘𝐴) + (exp‘-𝐴))) |
30 | 29 | oveq1d 7420 | . 2 ⊢ (𝐴 ∈ ℂ → (((exp‘(i · (i · 𝐴))) + (exp‘(-i · (i · 𝐴)))) / 2) = (((exp‘𝐴) + (exp‘-𝐴)) / 2)) |
31 | 5, 30 | eqtrd 2766 | 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 6537 (class class class)co 7405 ℂcc 11110 1c1 11113 ici 11114 + caddc 11115 · cmul 11117 -cneg 11449 / cdiv 11875 2c2 12271 expce 16011 cosccos 16014 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-ico 13336 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-fac 14239 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-ef 16017 df-cos 16020 |
This theorem is referenced by: rpcoshcl 16107 tanhlt1 16110 sinhpcosh 48056 |
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