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Theorem sinhpcosh 48283
Description: Prove that (sinhβ€˜π΄) + (coshβ€˜π΄) = (expβ€˜π΄) using the conventional hyperbolic trigonometric functions. (Contributed by David A. Wheeler, 27-May-2015.)
Assertion
Ref Expression
sinhpcosh (𝐴 ∈ β„‚ β†’ ((sinhβ€˜π΄) + (coshβ€˜π΄)) = (expβ€˜π΄))

Proof of Theorem sinhpcosh
StepHypRef Expression
1 sinhval-named 48279 . . . . 5 (𝐴 ∈ β„‚ β†’ (sinhβ€˜π΄) = ((sinβ€˜(i Β· 𝐴)) / i))
2 sinhval 16130 . . . . 5 (𝐴 ∈ β„‚ β†’ ((sinβ€˜(i Β· 𝐴)) / i) = (((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) / 2))
31, 2eqtrd 2765 . . . 4 (𝐴 ∈ β„‚ β†’ (sinhβ€˜π΄) = (((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) / 2))
4 coshval-named 48280 . . . . 5 (𝐴 ∈ β„‚ β†’ (coshβ€˜π΄) = (cosβ€˜(i Β· 𝐴)))
5 coshval 16131 . . . . 5 (𝐴 ∈ β„‚ β†’ (cosβ€˜(i Β· 𝐴)) = (((expβ€˜π΄) + (expβ€˜-𝐴)) / 2))
64, 5eqtrd 2765 . . . 4 (𝐴 ∈ β„‚ β†’ (coshβ€˜π΄) = (((expβ€˜π΄) + (expβ€˜-𝐴)) / 2))
73, 6oveq12d 7434 . . 3 (𝐴 ∈ β„‚ β†’ ((sinhβ€˜π΄) + (coshβ€˜π΄)) = ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) / 2) + (((expβ€˜π΄) + (expβ€˜-𝐴)) / 2)))
8 2cn 12317 . . . 4 2 ∈ β„‚
9 2ne0 12346 . . . 4 2 β‰  0
10 efcl 16058 . . . . . . 7 (𝐴 ∈ β„‚ β†’ (expβ€˜π΄) ∈ β„‚)
11 negcl 11490 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ -𝐴 ∈ β„‚)
12 efcl 16058 . . . . . . . 8 (-𝐴 ∈ β„‚ β†’ (expβ€˜-𝐴) ∈ β„‚)
1311, 12syl 17 . . . . . . 7 (𝐴 ∈ β„‚ β†’ (expβ€˜-𝐴) ∈ β„‚)
1410, 13addcld 11263 . . . . . 6 (𝐴 ∈ β„‚ β†’ ((expβ€˜π΄) + (expβ€˜-𝐴)) ∈ β„‚)
1510, 13subcld 11601 . . . . . . 7 (𝐴 ∈ β„‚ β†’ ((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) ∈ β„‚)
16 divdir 11927 . . . . . . 7 ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) ∈ β„‚ ∧ ((expβ€˜π΄) + (expβ€˜-𝐴)) ∈ β„‚ ∧ (2 ∈ β„‚ ∧ 2 β‰  0)) β†’ ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + ((expβ€˜π΄) + (expβ€˜-𝐴))) / 2) = ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) / 2) + (((expβ€˜π΄) + (expβ€˜-𝐴)) / 2)))
1715, 16syl3an1 1160 . . . . . 6 ((𝐴 ∈ β„‚ ∧ ((expβ€˜π΄) + (expβ€˜-𝐴)) ∈ β„‚ ∧ (2 ∈ β„‚ ∧ 2 β‰  0)) β†’ ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + ((expβ€˜π΄) + (expβ€˜-𝐴))) / 2) = ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) / 2) + (((expβ€˜π΄) + (expβ€˜-𝐴)) / 2)))
1814, 17syl3an2 1161 . . . . 5 ((𝐴 ∈ β„‚ ∧ 𝐴 ∈ β„‚ ∧ (2 ∈ β„‚ ∧ 2 β‰  0)) β†’ ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + ((expβ€˜π΄) + (expβ€˜-𝐴))) / 2) = ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) / 2) + (((expβ€˜π΄) + (expβ€˜-𝐴)) / 2)))
19183anidm12 1416 . . . 4 ((𝐴 ∈ β„‚ ∧ (2 ∈ β„‚ ∧ 2 β‰  0)) β†’ ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + ((expβ€˜π΄) + (expβ€˜-𝐴))) / 2) = ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) / 2) + (((expβ€˜π΄) + (expβ€˜-𝐴)) / 2)))
208, 9, 19mpanr12 703 . . 3 (𝐴 ∈ β„‚ β†’ ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + ((expβ€˜π΄) + (expβ€˜-𝐴))) / 2) = ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) / 2) + (((expβ€˜π΄) + (expβ€˜-𝐴)) / 2)))
21102timesd 12485 . . . . 5 (𝐴 ∈ β„‚ β†’ (2 Β· (expβ€˜π΄)) = ((expβ€˜π΄) + (expβ€˜π΄)))
2210, 13, 10nppcand 11626 . . . . 5 (𝐴 ∈ β„‚ β†’ ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + (expβ€˜π΄)) + (expβ€˜-𝐴)) = ((expβ€˜π΄) + (expβ€˜π΄)))
2315, 10, 13addassd 11266 . . . . 5 (𝐴 ∈ β„‚ β†’ ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + (expβ€˜π΄)) + (expβ€˜-𝐴)) = (((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + ((expβ€˜π΄) + (expβ€˜-𝐴))))
2421, 22, 233eqtr2rd 2772 . . . 4 (𝐴 ∈ β„‚ β†’ (((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + ((expβ€˜π΄) + (expβ€˜-𝐴))) = (2 Β· (expβ€˜π΄)))
2524oveq1d 7431 . . 3 (𝐴 ∈ β„‚ β†’ ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + ((expβ€˜π΄) + (expβ€˜-𝐴))) / 2) = ((2 Β· (expβ€˜π΄)) / 2))
267, 20, 253eqtr2d 2771 . 2 (𝐴 ∈ β„‚ β†’ ((sinhβ€˜π΄) + (coshβ€˜π΄)) = ((2 Β· (expβ€˜π΄)) / 2))
278a1i 11 . . 3 (𝐴 ∈ β„‚ β†’ 2 ∈ β„‚)
289a1i 11 . . 3 (𝐴 ∈ β„‚ β†’ 2 β‰  0)
2910, 27, 28divcan3d 12025 . 2 (𝐴 ∈ β„‚ β†’ ((2 Β· (expβ€˜π΄)) / 2) = (expβ€˜π΄))
3026, 29eqtrd 2765 1 (𝐴 ∈ β„‚ β†’ ((sinhβ€˜π΄) + (coshβ€˜π΄)) = (expβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  β€˜cfv 6543  (class class class)co 7416  β„‚cc 11136  0cc0 11138  ici 11140   + caddc 11141   Β· cmul 11143   βˆ’ cmin 11474  -cneg 11475   / cdiv 11901  2c2 12297  expce 16037  sincsin 16039  cosccos 16040  sinhcsinh 48273  coshccosh 48274
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-pm 8846  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-sup 9465  df-inf 9466  df-oi 9533  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-n0 12503  df-z 12589  df-uz 12853  df-rp 13007  df-ico 13362  df-fz 13517  df-fzo 13660  df-fl 13789  df-seq 13999  df-exp 14059  df-fac 14265  df-hash 14322  df-shft 15046  df-cj 15078  df-re 15079  df-im 15080  df-sqrt 15214  df-abs 15215  df-limsup 15447  df-clim 15464  df-rlim 15465  df-sum 15665  df-ef 16043  df-sin 16045  df-cos 16046  df-sinh 48276  df-cosh 48277
This theorem is referenced by: (None)
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