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Theorem sinhpcosh 32092
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 32088 . . . . 5  |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( sin `  (
_i  x.  A )
)  /  _i ) )
2 sinhval 13741 . . . . 5  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
31, 2eqtrd 2503 . . . 4  |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )
4 coshval-named 32089 . . . . 5  |-  ( A  e.  CC  ->  (cosh `  A )  =  ( cos `  ( _i  x.  A ) ) )
5 coshval 13742 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  ( _i  x.  A ) )  =  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) )
64, 5eqtrd 2503 . . . 4  |-  ( A  e.  CC  ->  (cosh `  A )  =  ( ( ( exp `  A
)  +  ( exp `  -u A ) )  /  2 ) )
73, 6oveq12d 6295 . . 3  |-  ( A  e.  CC  ->  (
(sinh `  A )  +  (cosh `  A )
)  =  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  +  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
8 2cn 10597 . . . 4  |-  2  e.  CC
9 2ne0 10619 . . . 4  |-  2  =/=  0
10 efcl 13671 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
11 negcl 9811 . . . . . . . 8  |-  ( A  e.  CC  ->  -u A  e.  CC )
12 efcl 13671 . . . . . . . 8  |-  ( -u A  e.  CC  ->  ( exp `  -u A
)  e.  CC )
1311, 12syl 16 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  -u A )  e.  CC )
1410, 13addcld 9606 . . . . . 6  |-  ( A  e.  CC  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  CC )
1510, 13subcld 9921 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
16 divdir 10221 . . . . . . 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 1256 . . . . . 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 1257 . . . . 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 1280 . . . 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 685 . . 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 10772 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( exp `  A ) )  =  ( ( exp `  A
)  +  ( exp `  A ) ) )
2210, 13, 10nppcand 9946 . . . . 5  |-  ( A  e.  CC  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  +  ( exp `  A
) )  +  ( exp `  -u A
) )  =  ( ( exp `  A
)  +  ( exp `  A ) ) )
2315, 10, 13addassd 9609 . . . . 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 2510 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  =  ( 2  x.  ( exp `  A ) ) )
2524oveq1d 6292 . . 3  |-  ( A  e.  CC  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  / 
2 )  =  ( ( 2  x.  ( exp `  A ) )  /  2 ) )
267, 20, 253eqtr2d 2509 . 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 10316 . 2  |-  ( A  e.  CC  ->  (
( 2  x.  ( exp `  A ) )  /  2 )  =  ( exp `  A
) )
3026, 29eqtrd 2503 1  |-  ( A  e.  CC  ->  (
(sinh `  A )  +  (cosh `  A )
)  =  ( exp `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657   ` cfv 5581  (class class class)co 6277   CCcc 9481   0cc0 9483   _ici 9485    + caddc 9486    x. cmul 9488    - cmin 9796   -ucneg 9797    / cdiv 10197   2c2 10576   expce 13650   sincsin 13652   cosccos 13653  sinhcsinh 32082  coshccosh 32083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-oi 7926  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-rp 11212  df-ico 11526  df-fz 11664  df-fzo 11784  df-fl 11888  df-seq 12066  df-exp 12125  df-fac 12311  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-sum 13460  df-ef 13656  df-sin 13658  df-cos 13659  df-sinh 32085  df-cosh 32086
This theorem is referenced by: (None)
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