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Theorem sinhpcosh 47271
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 47267 . . . . 5 (𝐴 ∈ β„‚ β†’ (sinhβ€˜π΄) = ((sinβ€˜(i Β· 𝐴)) / i))
2 sinhval 16041 . . . . 5 (𝐴 ∈ β„‚ β†’ ((sinβ€˜(i Β· 𝐴)) / i) = (((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) / 2))
31, 2eqtrd 2773 . . . 4 (𝐴 ∈ β„‚ β†’ (sinhβ€˜π΄) = (((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) / 2))
4 coshval-named 47268 . . . . 5 (𝐴 ∈ β„‚ β†’ (coshβ€˜π΄) = (cosβ€˜(i Β· 𝐴)))
5 coshval 16042 . . . . 5 (𝐴 ∈ β„‚ β†’ (cosβ€˜(i Β· 𝐴)) = (((expβ€˜π΄) + (expβ€˜-𝐴)) / 2))
64, 5eqtrd 2773 . . . 4 (𝐴 ∈ β„‚ β†’ (coshβ€˜π΄) = (((expβ€˜π΄) + (expβ€˜-𝐴)) / 2))
73, 6oveq12d 7376 . . 3 (𝐴 ∈ β„‚ β†’ ((sinhβ€˜π΄) + (coshβ€˜π΄)) = ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) / 2) + (((expβ€˜π΄) + (expβ€˜-𝐴)) / 2)))
8 2cn 12233 . . . 4 2 ∈ β„‚
9 2ne0 12262 . . . 4 2 β‰  0
10 efcl 15970 . . . . . . 7 (𝐴 ∈ β„‚ β†’ (expβ€˜π΄) ∈ β„‚)
11 negcl 11406 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ -𝐴 ∈ β„‚)
12 efcl 15970 . . . . . . . 8 (-𝐴 ∈ β„‚ β†’ (expβ€˜-𝐴) ∈ β„‚)
1311, 12syl 17 . . . . . . 7 (𝐴 ∈ β„‚ β†’ (expβ€˜-𝐴) ∈ β„‚)
1410, 13addcld 11179 . . . . . 6 (𝐴 ∈ β„‚ β†’ ((expβ€˜π΄) + (expβ€˜-𝐴)) ∈ β„‚)
1510, 13subcld 11517 . . . . . . 7 (𝐴 ∈ β„‚ β†’ ((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) ∈ β„‚)
16 divdir 11843 . . . . . . 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 1164 . . . . . 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 1165 . . . . 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 1420 . . . 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 704 . . 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 12401 . . . . 5 (𝐴 ∈ β„‚ β†’ (2 Β· (expβ€˜π΄)) = ((expβ€˜π΄) + (expβ€˜π΄)))
2210, 13, 10nppcand 11542 . . . . 5 (𝐴 ∈ β„‚ β†’ ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + (expβ€˜π΄)) + (expβ€˜-𝐴)) = ((expβ€˜π΄) + (expβ€˜π΄)))
2315, 10, 13addassd 11182 . . . . 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 2780 . . . 4 (𝐴 ∈ β„‚ β†’ (((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + ((expβ€˜π΄) + (expβ€˜-𝐴))) = (2 Β· (expβ€˜π΄)))
2524oveq1d 7373 . . 3 (𝐴 ∈ β„‚ β†’ ((((expβ€˜π΄) βˆ’ (expβ€˜-𝐴)) + ((expβ€˜π΄) + (expβ€˜-𝐴))) / 2) = ((2 Β· (expβ€˜π΄)) / 2))
267, 20, 253eqtr2d 2779 . 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 11941 . 2 (𝐴 ∈ β„‚ β†’ ((2 Β· (expβ€˜π΄)) / 2) = (expβ€˜π΄))
3026, 29eqtrd 2773 1 (𝐴 ∈ β„‚ β†’ ((sinhβ€˜π΄) + (coshβ€˜π΄)) = (expβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054  0cc0 11056  ici 11058   + caddc 11059   Β· cmul 11061   βˆ’ cmin 11390  -cneg 11391   / cdiv 11817  2c2 12213  expce 15949  sincsin 15951  cosccos 15952  sinhcsinh 47261  coshccosh 47262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-pm 8771  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-inf 9384  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-rp 12921  df-ico 13276  df-fz 13431  df-fzo 13574  df-fl 13703  df-seq 13913  df-exp 13974  df-fac 14180  df-hash 14237  df-shft 14958  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-limsup 15359  df-clim 15376  df-rlim 15377  df-sum 15577  df-ef 15955  df-sin 15957  df-cos 15958  df-sinh 47264  df-cosh 47265
This theorem is referenced by: (None)
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