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Theorem sinhpcosh 38780
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  e.  CC  ->  (
(sinh `  A )  +  (cosh `  A )
)  =  ( exp `  A ) )

Proof of Theorem sinhpcosh
StepHypRef Expression
1 sinhval-named 38776 . . . . 5  |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( sin `  (
_i  x.  A )
)  /  _i ) )
2 sinhval 14098 . . . . 5  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
31, 2eqtrd 2443 . . . 4  |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )
4 coshval-named 38777 . . . . 5  |-  ( A  e.  CC  ->  (cosh `  A )  =  ( cos `  ( _i  x.  A ) ) )
5 coshval 14099 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  ( _i  x.  A ) )  =  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) )
64, 5eqtrd 2443 . . . 4  |-  ( A  e.  CC  ->  (cosh `  A )  =  ( ( ( exp `  A
)  +  ( exp `  -u A ) )  /  2 ) )
73, 6oveq12d 6296 . . 3  |-  ( A  e.  CC  ->  (
(sinh `  A )  +  (cosh `  A )
)  =  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  +  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
8 2cn 10647 . . . 4  |-  2  e.  CC
9 2ne0 10669 . . . 4  |-  2  =/=  0
10 efcl 14027 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
11 negcl 9856 . . . . . . . 8  |-  ( A  e.  CC  ->  -u A  e.  CC )
12 efcl 14027 . . . . . . . 8  |-  ( -u A  e.  CC  ->  ( exp `  -u A
)  e.  CC )
1311, 12syl 17 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  -u A )  e.  CC )
1410, 13addcld 9645 . . . . . 6  |-  ( A  e.  CC  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  CC )
1510, 13subcld 9967 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
16 divdir 10271 . . . . . . 7  |-  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  e.  CC  /\  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  CC  /\  (
2  e.  CC  /\  2  =/=  0 ) )  ->  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  / 
2 )  =  ( ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 )  +  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
1715, 16syl3an1 1263 . . . . . 6  |-  ( ( A  e.  CC  /\  ( ( exp `  A
)  +  ( exp `  -u A ) )  e.  CC  /\  (
2  e.  CC  /\  2  =/=  0 ) )  ->  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  / 
2 )  =  ( ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 )  +  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
1814, 17syl3an2 1264 . . . . 5  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  (
2  e.  CC  /\  2  =/=  0 ) )  ->  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  / 
2 )  =  ( ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 )  +  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
19183anidm12 1287 . . . 4  |-  ( ( A  e.  CC  /\  ( 2  e.  CC  /\  2  =/=  0 ) )  ->  ( (
( ( exp `  A
)  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  / 
2 )  =  ( ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 )  +  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
208, 9, 19mpanr12 683 . . 3  |-  ( A  e.  CC  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  / 
2 )  =  ( ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 )  +  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
21102timesd 10822 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( exp `  A ) )  =  ( ( exp `  A
)  +  ( exp `  A ) ) )
2210, 13, 10nppcand 9992 . . . . 5  |-  ( A  e.  CC  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  +  ( exp `  A
) )  +  ( exp `  -u A
) )  =  ( ( exp `  A
)  +  ( exp `  A ) ) )
2315, 10, 13addassd 9648 . . . . 5  |-  ( A  e.  CC  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  +  ( exp `  A
) )  +  ( exp `  -u A
) )  =  ( ( ( exp `  A
)  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) ) )
2421, 22, 233eqtr2rd 2450 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  =  ( 2  x.  ( exp `  A ) ) )
2524oveq1d 6293 . . 3  |-  ( A  e.  CC  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  / 
2 )  =  ( ( 2  x.  ( exp `  A ) )  /  2 ) )
267, 20, 253eqtr2d 2449 . 2  |-  ( A  e.  CC  ->  (
(sinh `  A )  +  (cosh `  A )
)  =  ( ( 2  x.  ( exp `  A ) )  / 
2 ) )
278a1i 11 . . 3  |-  ( A  e.  CC  ->  2  e.  CC )
289a1i 11 . . 3  |-  ( A  e.  CC  ->  2  =/=  0 )
2910, 27, 28divcan3d 10366 . 2  |-  ( A  e.  CC  ->  (
( 2  x.  ( exp `  A ) )  /  2 )  =  ( exp `  A
) )
3026, 29eqtrd 2443 1  |-  ( A  e.  CC  ->  (
(sinh `  A )  +  (cosh `  A )
)  =  ( exp `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   ` cfv 5569  (class class class)co 6278   CCcc 9520   0cc0 9522   _ici 9524    + caddc 9525    x. cmul 9527    - cmin 9841   -ucneg 9842    / cdiv 10247   2c2 10626   expce 14006   sincsin 14008   cosccos 14009  sinhcsinh 38770  coshccosh 38771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-ico 11588  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-fac 12398  df-hash 12453  df-shft 13049  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-limsup 13443  df-clim 13460  df-rlim 13461  df-sum 13658  df-ef 14012  df-sin 14014  df-cos 14015  df-sinh 38773  df-cosh 38774
This theorem is referenced by: (None)
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