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Theorem sinhpcosh 31073
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 31069 . . . . 5  |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( sin `  (
_i  x.  A )
)  /  _i ) )
2 sinhval 13437 . . . . 5  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
31, 2eqtrd 2474 . . . 4  |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )
4 coshval-named 31070 . . . . 5  |-  ( A  e.  CC  ->  (cosh `  A )  =  ( cos `  ( _i  x.  A ) ) )
5 coshval 13438 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  ( _i  x.  A ) )  =  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) )
64, 5eqtrd 2474 . . . 4  |-  ( A  e.  CC  ->  (cosh `  A )  =  ( ( ( exp `  A
)  +  ( exp `  -u A ) )  /  2 ) )
73, 6oveq12d 6108 . . 3  |-  ( A  e.  CC  ->  (
(sinh `  A )  +  (cosh `  A )
)  =  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  +  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
8 2cn 10391 . . . 4  |-  2  e.  CC
9 2ne0 10413 . . . 4  |-  2  =/=  0
10 efcl 13367 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
11 negcl 9609 . . . . . . . 8  |-  ( A  e.  CC  ->  -u A  e.  CC )
12 efcl 13367 . . . . . . . 8  |-  ( -u A  e.  CC  ->  ( exp `  -u A
)  e.  CC )
1311, 12syl 16 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  -u A )  e.  CC )
1410, 13addcld 9404 . . . . . 6  |-  ( A  e.  CC  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  CC )
1510, 13subcld 9718 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
16 divdir 10016 . . . . . . 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 1251 . . . . . 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 1252 . . . . 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 1275 . . . 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 10566 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( exp `  A ) )  =  ( ( exp `  A
)  +  ( exp `  A ) ) )
2210, 13, 10nppcand 9743 . . . . 5  |-  ( A  e.  CC  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  +  ( exp `  A
) )  +  ( exp `  -u A
) )  =  ( ( exp `  A
)  +  ( exp `  A ) ) )
2315, 10, 13addassd 9407 . . . . 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 2481 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  =  ( 2  x.  ( exp `  A ) ) )
2524oveq1d 6105 . . 3  |-  ( A  e.  CC  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  / 
2 )  =  ( ( 2  x.  ( exp `  A ) )  /  2 ) )
267, 20, 253eqtr2d 2480 . 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 10111 . 2  |-  ( A  e.  CC  ->  (
( 2  x.  ( exp `  A ) )  /  2 )  =  ( exp `  A
) )
3026, 29eqtrd 2474 1  |-  ( A  e.  CC  ->  (
(sinh `  A )  +  (cosh `  A )
)  =  ( exp `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605   ` cfv 5417  (class class class)co 6090   CCcc 9279   0cc0 9281   _ici 9283    + caddc 9284    x. cmul 9286    - cmin 9594   -ucneg 9595    / cdiv 9992   2c2 10370   expce 13346   sincsin 13348   cosccos 13349  sinhcsinh 31063  coshccosh 31064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-ico 11305  df-fz 11437  df-fzo 11548  df-fl 11641  df-seq 11806  df-exp 11865  df-fac 12051  df-hash 12103  df-shft 12555  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-limsup 12948  df-clim 12965  df-rlim 12966  df-sum 13163  df-ef 13352  df-sin 13354  df-cos 13355  df-sinh 31066  df-cosh 31067
This theorem is referenced by: (None)
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