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Mirrors > Home > MPE Home > Th. List > Mathboxes > sinhpcosh | Structured version Visualization version GIF version |
Description: Prove that (sinh‘𝐴) + (cosh‘𝐴) = (exp‘𝐴) using the conventional hyperbolic trigonometric functions. (Contributed by David A. Wheeler, 27-May-2015.) |
Ref | Expression |
---|---|
sinhpcosh | ⊢ (𝐴 ∈ ℂ → ((sinh‘𝐴) + (cosh‘𝐴)) = (exp‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sinhval-named 44842 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i)) | |
2 | sinhval 15510 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((sin‘(i · 𝐴)) / i) = (((exp‘𝐴) − (exp‘-𝐴)) / 2)) | |
3 | 1, 2 | eqtrd 2859 | . . . 4 ⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) = (((exp‘𝐴) − (exp‘-𝐴)) / 2)) |
4 | coshval-named 44843 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴))) | |
5 | coshval 15511 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘(i · 𝐴)) = (((exp‘𝐴) + (exp‘-𝐴)) / 2)) | |
6 | 4, 5 | eqtrd 2859 | . . . 4 ⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) = (((exp‘𝐴) + (exp‘-𝐴)) / 2)) |
7 | 3, 6 | oveq12d 7177 | . . 3 ⊢ (𝐴 ∈ ℂ → ((sinh‘𝐴) + (cosh‘𝐴)) = ((((exp‘𝐴) − (exp‘-𝐴)) / 2) + (((exp‘𝐴) + (exp‘-𝐴)) / 2))) |
8 | 2cn 11715 | . . . 4 ⊢ 2 ∈ ℂ | |
9 | 2ne0 11744 | . . . 4 ⊢ 2 ≠ 0 | |
10 | efcl 15439 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) | |
11 | negcl 10889 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
12 | efcl 15439 | . . . . . . . 8 ⊢ (-𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) | |
13 | 11, 12 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ) |
14 | 10, 13 | addcld 10663 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((exp‘𝐴) + (exp‘-𝐴)) ∈ ℂ) |
15 | 10, 13 | subcld 11000 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((exp‘𝐴) − (exp‘-𝐴)) ∈ ℂ) |
16 | divdir 11326 | . . . . . . 7 ⊢ ((((exp‘𝐴) − (exp‘-𝐴)) ∈ ℂ ∧ ((exp‘𝐴) + (exp‘-𝐴)) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((((exp‘𝐴) − (exp‘-𝐴)) + ((exp‘𝐴) + (exp‘-𝐴))) / 2) = ((((exp‘𝐴) − (exp‘-𝐴)) / 2) + (((exp‘𝐴) + (exp‘-𝐴)) / 2))) | |
17 | 15, 16 | syl3an1 1159 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((exp‘𝐴) + (exp‘-𝐴)) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((((exp‘𝐴) − (exp‘-𝐴)) + ((exp‘𝐴) + (exp‘-𝐴))) / 2) = ((((exp‘𝐴) − (exp‘-𝐴)) / 2) + (((exp‘𝐴) + (exp‘-𝐴)) / 2))) |
18 | 14, 17 | syl3an2 1160 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((((exp‘𝐴) − (exp‘-𝐴)) + ((exp‘𝐴) + (exp‘-𝐴))) / 2) = ((((exp‘𝐴) − (exp‘-𝐴)) / 2) + (((exp‘𝐴) + (exp‘-𝐴)) / 2))) |
19 | 18 | 3anidm12 1415 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((((exp‘𝐴) − (exp‘-𝐴)) + ((exp‘𝐴) + (exp‘-𝐴))) / 2) = ((((exp‘𝐴) − (exp‘-𝐴)) / 2) + (((exp‘𝐴) + (exp‘-𝐴)) / 2))) |
20 | 8, 9, 19 | mpanr12 703 | . . 3 ⊢ (𝐴 ∈ ℂ → ((((exp‘𝐴) − (exp‘-𝐴)) + ((exp‘𝐴) + (exp‘-𝐴))) / 2) = ((((exp‘𝐴) − (exp‘-𝐴)) / 2) + (((exp‘𝐴) + (exp‘-𝐴)) / 2))) |
21 | 10 | 2timesd 11883 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (2 · (exp‘𝐴)) = ((exp‘𝐴) + (exp‘𝐴))) |
22 | 10, 13, 10 | nppcand 11025 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((((exp‘𝐴) − (exp‘-𝐴)) + (exp‘𝐴)) + (exp‘-𝐴)) = ((exp‘𝐴) + (exp‘𝐴))) |
23 | 15, 10, 13 | addassd 10666 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((((exp‘𝐴) − (exp‘-𝐴)) + (exp‘𝐴)) + (exp‘-𝐴)) = (((exp‘𝐴) − (exp‘-𝐴)) + ((exp‘𝐴) + (exp‘-𝐴)))) |
24 | 21, 22, 23 | 3eqtr2rd 2866 | . . . 4 ⊢ (𝐴 ∈ ℂ → (((exp‘𝐴) − (exp‘-𝐴)) + ((exp‘𝐴) + (exp‘-𝐴))) = (2 · (exp‘𝐴))) |
25 | 24 | oveq1d 7174 | . . 3 ⊢ (𝐴 ∈ ℂ → ((((exp‘𝐴) − (exp‘-𝐴)) + ((exp‘𝐴) + (exp‘-𝐴))) / 2) = ((2 · (exp‘𝐴)) / 2)) |
26 | 7, 20, 25 | 3eqtr2d 2865 | . 2 ⊢ (𝐴 ∈ ℂ → ((sinh‘𝐴) + (cosh‘𝐴)) = ((2 · (exp‘𝐴)) / 2)) |
27 | 8 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → 2 ∈ ℂ) |
28 | 9 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → 2 ≠ 0) |
29 | 10, 27, 28 | divcan3d 11424 | . 2 ⊢ (𝐴 ∈ ℂ → ((2 · (exp‘𝐴)) / 2) = (exp‘𝐴)) |
30 | 26, 29 | eqtrd 2859 | 1 ⊢ (𝐴 ∈ ℂ → ((sinh‘𝐴) + (cosh‘𝐴)) = (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 0cc0 10540 ici 10542 + caddc 10543 · cmul 10545 − cmin 10873 -cneg 10874 / cdiv 11300 2c2 11695 expce 15418 sincsin 15420 cosccos 15421 sinhcsinh 44836 coshccosh 44837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-fac 13637 df-hash 13694 df-shft 14429 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 df-rlim 14849 df-sum 15046 df-ef 15424 df-sin 15426 df-cos 15427 df-sinh 44839 df-cosh 44840 |
This theorem is referenced by: (None) |
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