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Theorem tanhlt1 12761
Description: The hyperbolic tangent of a real number is upper bounded by  1. (Contributed by Mario Carneiro, 4-Apr-2015.)
Assertion
Ref Expression
tanhlt1  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  <  1 )

Proof of Theorem tanhlt1
StepHypRef Expression
1 ax-icn 9049 . . . . . . 7  |-  _i  e.  CC
2 recn 9080 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
3 mulcl 9074 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
41, 2, 3sylancr 645 . . . . . 6  |-  ( A  e.  RR  ->  (
_i  x.  A )  e.  CC )
5 rpcoshcl 12758 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR+ )
65rpne0d 10653 . . . . . 6  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  =/=  0 )
7 tanval 12729 . . . . . 6  |-  ( ( ( _i  x.  A
)  e.  CC  /\  ( cos `  ( _i  x.  A ) )  =/=  0 )  -> 
( tan `  (
_i  x.  A )
)  =  ( ( sin `  ( _i  x.  A ) )  /  ( cos `  (
_i  x.  A )
) ) )
84, 6, 7syl2anc 643 . . . . 5  |-  ( A  e.  RR  ->  ( tan `  ( _i  x.  A ) )  =  ( ( sin `  (
_i  x.  A )
)  /  ( cos `  ( _i  x.  A
) ) ) )
98oveq1d 6096 . . . 4  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( ( sin `  ( _i  x.  A ) )  /  ( cos `  (
_i  x.  A )
) )  /  _i ) )
104sincld 12731 . . . . 5  |-  ( A  e.  RR  ->  ( sin `  ( _i  x.  A ) )  e.  CC )
11 recoshcl 12759 . . . . . 6  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR )
1211recnd 9114 . . . . 5  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  CC )
131a1i 11 . . . . 5  |-  ( A  e.  RR  ->  _i  e.  CC )
14 ine0 9469 . . . . . 6  |-  _i  =/=  0
1514a1i 11 . . . . 5  |-  ( A  e.  RR  ->  _i  =/=  0 )
1610, 12, 13, 6, 15divdiv32d 9815 . . . 4  |-  ( A  e.  RR  ->  (
( ( sin `  (
_i  x.  A )
)  /  ( cos `  ( _i  x.  A
) ) )  /  _i )  =  (
( ( sin `  (
_i  x.  A )
)  /  _i )  /  ( cos `  (
_i  x.  A )
) ) )
17 sinhval 12755 . . . . . 6  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
182, 17syl 16 . . . . 5  |-  ( A  e.  RR  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
19 coshval 12756 . . . . . 6  |-  ( A  e.  CC  ->  ( cos `  ( _i  x.  A ) )  =  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) )
202, 19syl 16 . . . . 5  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  =  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) )
2118, 20oveq12d 6099 . . . 4  |-  ( A  e.  RR  ->  (
( ( sin `  (
_i  x.  A )
)  /  _i )  /  ( cos `  (
_i  x.  A )
) )  =  ( ( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 )  / 
( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
229, 16, 213eqtrd 2472 . . 3  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  / 
( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
23 reefcl 12689 . . . . . 6  |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR )
24 renegcl 9364 . . . . . . 7  |-  ( A  e.  RR  ->  -u A  e.  RR )
2524reefcld 12690 . . . . . 6  |-  ( A  e.  RR  ->  ( exp `  -u A )  e.  RR )
2623, 25resubcld 9465 . . . . 5  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  RR )
2726recnd 9114 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
2823, 25readdcld 9115 . . . . 5  |-  ( A  e.  RR  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  RR )
2928recnd 9114 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  CC )
30 2cn 10070 . . . . 5  |-  2  e.  CC
3130a1i 11 . . . 4  |-  ( A  e.  RR  ->  2  e.  CC )
3220, 6eqnetrrd 2621 . . . . 5  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  +  ( exp `  -u A ) )  /  2 )  =/=  0 )
33 2ne0 10083 . . . . . . 7  |-  2  =/=  0
3433a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  2  =/=  0 )
3529, 31, 34divne0bd 9802 . . . . 5  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  +  ( exp `  -u A ) )  =/=  0  <->  ( (
( exp `  A
)  +  ( exp `  -u A ) )  /  2 )  =/=  0 ) )
3632, 35mpbird 224 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  =/=  0 )
3727, 29, 31, 36, 34divcan7d 9818 . . 3  |-  ( A  e.  RR  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  /  2 )  / 
( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) )  =  ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  ( ( exp `  A )  +  ( exp `  -u A
) ) ) )
3822, 37eqtrd 2468 . 2  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  (
( exp `  A
)  +  ( exp `  -u A ) ) ) )
3924rpefcld 12706 . . . . . 6  |-  ( A  e.  RR  ->  ( exp `  -u A )  e.  RR+ )
4023, 39ltsubrpd 10676 . . . . 5  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  <  ( exp `  A
) )
4123, 39ltaddrpd 10677 . . . . 5  |-  ( A  e.  RR  ->  ( exp `  A )  < 
( ( exp `  A
)  +  ( exp `  -u A ) ) )
4226, 23, 28, 40, 41lttrd 9231 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  <  ( ( exp `  A )  +  ( exp `  -u A
) ) )
4329mulid1d 9105 . . . 4  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  +  ( exp `  -u A ) )  x.  1 )  =  ( ( exp `  A
)  +  ( exp `  -u A ) ) )
4442, 43breqtrrd 4238 . . 3  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  <  ( ( ( exp `  A )  +  ( exp `  -u A
) )  x.  1 ) )
45 1re 9090 . . . . 5  |-  1  e.  RR
4645a1i 11 . . . 4  |-  ( A  e.  RR  ->  1  e.  RR )
47 efgt0 12704 . . . . 5  |-  ( A  e.  RR  ->  0  <  ( exp `  A
) )
48 efgt0 12704 . . . . . 6  |-  ( -u A  e.  RR  ->  0  <  ( exp `  -u A
) )
4924, 48syl 16 . . . . 5  |-  ( A  e.  RR  ->  0  <  ( exp `  -u A
) )
5023, 25, 47, 49addgt0d 9601 . . . 4  |-  ( A  e.  RR  ->  0  <  ( ( exp `  A
)  +  ( exp `  -u A ) ) )
51 ltdivmul 9882 . . . 4  |-  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  e.  RR  /\  1  e.  RR  /\  ( ( ( exp `  A
)  +  ( exp `  -u A ) )  e.  RR  /\  0  <  ( ( exp `  A
)  +  ( exp `  -u A ) ) ) )  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  /  ( ( exp `  A )  +  ( exp `  -u A
) ) )  <  1  <->  ( ( exp `  A )  -  ( exp `  -u A ) )  <  ( ( ( exp `  A )  +  ( exp `  -u A
) )  x.  1 ) ) )
5226, 46, 28, 50, 51syl112anc 1188 . . 3  |-  ( A  e.  RR  ->  (
( ( ( exp `  A )  -  ( exp `  -u A ) )  /  ( ( exp `  A )  +  ( exp `  -u A
) ) )  <  1  <->  ( ( exp `  A )  -  ( exp `  -u A ) )  <  ( ( ( exp `  A )  +  ( exp `  -u A
) )  x.  1 ) ) )
5344, 52mpbird 224 . 2  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  ( ( exp `  A )  +  ( exp `  -u A
) ) )  <  1 )
5438, 53eqbrtrd 4232 1  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  <  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991   _ici 8992    + caddc 8993    x. cmul 8995    < clt 9120    - cmin 9291   -ucneg 9292    / cdiv 9677   2c2 10049   expce 12664   sincsin 12666   cosccos 12667   tanctan 12668
This theorem is referenced by:  tanhbnd  12762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ico 10922  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672  df-cos 12673  df-tan 12674
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