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Theorem sinhpcosh 42280
Description: Prove that (sinh‘𝐴) + (cosh‘𝐴) = (exp‘𝐴) using the conventional hyperbolic trigonometric functions. (Contributed by David A. Wheeler, 27-May-2015.)
Assertion
Ref Expression
sinhpcosh (𝐴 ∈ ℂ → ((sinh‘𝐴) + (cosh‘𝐴)) = (exp‘𝐴))

Proof of Theorem sinhpcosh
StepHypRef Expression
1 sinhval-named 42276 . . . . 5 (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i))
2 sinhval 14723 . . . . 5 (𝐴 ∈ ℂ → ((sin‘(i · 𝐴)) / i) = (((exp‘𝐴) − (exp‘-𝐴)) / 2))
31, 2eqtrd 2644 . . . 4 (𝐴 ∈ ℂ → (sinh‘𝐴) = (((exp‘𝐴) − (exp‘-𝐴)) / 2))
4 coshval-named 42277 . . . . 5 (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴)))
5 coshval 14724 . . . . 5 (𝐴 ∈ ℂ → (cos‘(i · 𝐴)) = (((exp‘𝐴) + (exp‘-𝐴)) / 2))
64, 5eqtrd 2644 . . . 4 (𝐴 ∈ ℂ → (cosh‘𝐴) = (((exp‘𝐴) + (exp‘-𝐴)) / 2))
73, 6oveq12d 6567 . . 3 (𝐴 ∈ ℂ → ((sinh‘𝐴) + (cosh‘𝐴)) = ((((exp‘𝐴) − (exp‘-𝐴)) / 2) + (((exp‘𝐴) + (exp‘-𝐴)) / 2)))
8 2cn 10968 . . . 4 2 ∈ ℂ
9 2ne0 10990 . . . 4 2 ≠ 0
10 efcl 14652 . . . . . . 7 (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
11 negcl 10160 . . . . . . . 8 (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
12 efcl 14652 . . . . . . . 8 (-𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ)
1311, 12syl 17 . . . . . . 7 (𝐴 ∈ ℂ → (exp‘-𝐴) ∈ ℂ)
1410, 13addcld 9938 . . . . . 6 (𝐴 ∈ ℂ → ((exp‘𝐴) + (exp‘-𝐴)) ∈ ℂ)
1510, 13subcld 10271 . . . . . . 7 (𝐴 ∈ ℂ → ((exp‘𝐴) − (exp‘-𝐴)) ∈ ℂ)
16 divdir 10589 . . . . . . 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)))
1715, 16syl3an1 1351 . . . . . 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)))
1814, 17syl3an2 1352 . . . . 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)))
19183anidm12 1375 . . . 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)))
208, 9, 19mpanr12 717 . . 3 (𝐴 ∈ ℂ → ((((exp‘𝐴) − (exp‘-𝐴)) + ((exp‘𝐴) + (exp‘-𝐴))) / 2) = ((((exp‘𝐴) − (exp‘-𝐴)) / 2) + (((exp‘𝐴) + (exp‘-𝐴)) / 2)))
21102timesd 11152 . . . . 5 (𝐴 ∈ ℂ → (2 · (exp‘𝐴)) = ((exp‘𝐴) + (exp‘𝐴)))
2210, 13, 10nppcand 10296 . . . . 5 (𝐴 ∈ ℂ → ((((exp‘𝐴) − (exp‘-𝐴)) + (exp‘𝐴)) + (exp‘-𝐴)) = ((exp‘𝐴) + (exp‘𝐴)))
2315, 10, 13addassd 9941 . . . . 5 (𝐴 ∈ ℂ → ((((exp‘𝐴) − (exp‘-𝐴)) + (exp‘𝐴)) + (exp‘-𝐴)) = (((exp‘𝐴) − (exp‘-𝐴)) + ((exp‘𝐴) + (exp‘-𝐴))))
2421, 22, 233eqtr2rd 2651 . . . 4 (𝐴 ∈ ℂ → (((exp‘𝐴) − (exp‘-𝐴)) + ((exp‘𝐴) + (exp‘-𝐴))) = (2 · (exp‘𝐴)))
2524oveq1d 6564 . . 3 (𝐴 ∈ ℂ → ((((exp‘𝐴) − (exp‘-𝐴)) + ((exp‘𝐴) + (exp‘-𝐴))) / 2) = ((2 · (exp‘𝐴)) / 2))
267, 20, 253eqtr2d 2650 . 2 (𝐴 ∈ ℂ → ((sinh‘𝐴) + (cosh‘𝐴)) = ((2 · (exp‘𝐴)) / 2))
278a1i 11 . . 3 (𝐴 ∈ ℂ → 2 ∈ ℂ)
289a1i 11 . . 3 (𝐴 ∈ ℂ → 2 ≠ 0)
2910, 27, 28divcan3d 10685 . 2 (𝐴 ∈ ℂ → ((2 · (exp‘𝐴)) / 2) = (exp‘𝐴))
3026, 29eqtrd 2644 1 (𝐴 ∈ ℂ → ((sinh‘𝐴) + (cosh‘𝐴)) = (exp‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wne 2780  cfv 5804  (class class class)co 6549  cc 9813  0cc0 9815  ici 9817   + caddc 9818   · cmul 9820  cmin 10145  -cneg 10146   / cdiv 10563  2c2 10947  expce 14631  sincsin 14633  cosccos 14634  sinhcsinh 42270  coshccosh 42271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-fac 12923  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-sinh 42273  df-cosh 42274
This theorem is referenced by: (None)
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