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Theorem sinhpcosh 28484
Description: Prove that  (sinh `  A )  +  (cosh `  A )  =  ( exp `  A ) using the conventional hyperbolic trig 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 28480 . . . . 5  |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( sin `  (
_i  x.  A )
)  /  _i ) )
2 sinhval 12756 . . . . 5  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
31, 2eqtrd 2469 . . . 4  |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 ) )
4 coshval-named 28481 . . . . 5  |-  ( A  e.  CC  ->  (cosh `  A )  =  ( cos `  ( _i  x.  A ) ) )
5 coshval 12757 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  ( _i  x.  A ) )  =  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) )
64, 5eqtrd 2469 . . . 4  |-  ( A  e.  CC  ->  (cosh `  A )  =  ( ( ( exp `  A
)  +  ( exp `  -u A ) )  /  2 ) )
73, 6oveq12d 6100 . . 3  |-  ( A  e.  CC  ->  (
(sinh `  A )  +  (cosh `  A )
)  =  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  +  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
8 2cn 10071 . . . 4  |-  2  e.  CC
9 2ne0 10084 . . . 4  |-  2  =/=  0
10 efcl 12686 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
11 negcl 9307 . . . . . . . 8  |-  ( A  e.  CC  ->  -u A  e.  CC )
12 efcl 12686 . . . . . . . 8  |-  ( -u A  e.  CC  ->  ( exp `  -u A
)  e.  CC )
1311, 12syl 16 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  -u A )  e.  CC )
1410, 13addcld 9108 . . . . . 6  |-  ( A  e.  CC  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  CC )
1510, 13subcld 9412 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
16 divdir 9702 . . . . . . 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 1218 . . . . . 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 1219 . . . . 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 1242 . . . 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 668 . . 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 10211 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( exp `  A ) )  =  ( ( exp `  A
)  +  ( exp `  A ) ) )
2210, 13, 10nppcand 9437 . . . . 5  |-  ( A  e.  CC  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  +  ( exp `  A
) )  +  ( exp `  -u A
) )  =  ( ( exp `  A
)  +  ( exp `  A ) ) )
2315, 10, 13addassd 9111 . . . . 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 2476 . . . 4  |-  ( A  e.  CC  ->  (
( ( exp `  A
)  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  =  ( 2  x.  ( exp `  A ) ) )
2524oveq1d 6097 . . 3  |-  ( A  e.  CC  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  +  ( ( exp `  A )  +  ( exp `  -u A
) ) )  / 
2 )  =  ( ( 2  x.  ( exp `  A ) )  /  2 ) )
267, 20, 253eqtr2d 2475 . 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 9796 . 2  |-  ( A  e.  CC  ->  (
( 2  x.  ( exp `  A ) )  /  2 )  =  ( exp `  A
) )
3026, 29eqtrd 2469 1  |-  ( A  e.  CC  ->  (
(sinh `  A )  +  (cosh `  A )
)  =  ( exp `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   ` cfv 5455  (class class class)co 6082   CCcc 8989   0cc0 8991   _ici 8993    + caddc 8994    x. cmul 8996    - cmin 9292   -ucneg 9293    / cdiv 9678   2c2 10050   expce 12665   sincsin 12667   cosccos 12668  sinhcsinh 28474  coshccosh 28475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070  ax-mulf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-pm 7022  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-oi 7480  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-ico 10923  df-fz 11045  df-fzo 11137  df-fl 11203  df-seq 11325  df-exp 11384  df-fac 11568  df-hash 11620  df-shft 11883  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-limsup 12266  df-clim 12283  df-rlim 12284  df-sum 12481  df-ef 12671  df-sin 12673  df-cos 12674  df-sinh 28477  df-cosh 28478
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