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Theorem tanhlt1 12456
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 8812 . . . . . . 7  |-  _i  e.  CC
2 recn 8843 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
3 mulcl 8837 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
41, 2, 3sylancr 644 . . . . . 6  |-  ( A  e.  RR  ->  (
_i  x.  A )  e.  CC )
5 rpcoshcl 12453 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR+ )
65rpne0d 10411 . . . . . 6  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  =/=  0 )
7 tanval 12424 . . . . . 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 642 . . . . 5  |-  ( A  e.  RR  ->  ( tan `  ( _i  x.  A ) )  =  ( ( sin `  (
_i  x.  A )
)  /  ( cos `  ( _i  x.  A
) ) ) )
98oveq1d 5889 . . . 4  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( ( sin `  ( _i  x.  A ) )  /  ( cos `  (
_i  x.  A )
) )  /  _i ) )
104sincld 12426 . . . . 5  |-  ( A  e.  RR  ->  ( sin `  ( _i  x.  A ) )  e.  CC )
11 recoshcl 12454 . . . . . 6  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR )
1211recnd 8877 . . . . 5  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  CC )
131a1i 10 . . . . 5  |-  ( A  e.  RR  ->  _i  e.  CC )
14 ine0 9231 . . . . . 6  |-  _i  =/=  0
1514a1i 10 . . . . 5  |-  ( A  e.  RR  ->  _i  =/=  0 )
1610, 12, 13, 6, 15divdiv32d 9577 . . . 4  |-  ( A  e.  RR  ->  (
( ( sin `  (
_i  x.  A )
)  /  ( cos `  ( _i  x.  A
) ) )  /  _i )  =  (
( ( sin `  (
_i  x.  A )
)  /  _i )  /  ( cos `  (
_i  x.  A )
) ) )
17 sinhval 12450 . . . . . 6  |-  ( A  e.  CC  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
182, 17syl 15 . . . . 5  |-  ( A  e.  RR  ->  (
( sin `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  2
) )
19 coshval 12451 . . . . . 6  |-  ( A  e.  CC  ->  ( cos `  ( _i  x.  A ) )  =  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) )
202, 19syl 15 . . . . 5  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  =  ( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) )
2118, 20oveq12d 5892 . . . 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 2332 . . 3  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  / 
( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
23 reefcl 12384 . . . . . 6  |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR )
24 renegcl 9126 . . . . . . 7  |-  ( A  e.  RR  ->  -u A  e.  RR )
25 reefcl 12384 . . . . . . 7  |-  ( -u A  e.  RR  ->  ( exp `  -u A
)  e.  RR )
2624, 25syl 15 . . . . . 6  |-  ( A  e.  RR  ->  ( exp `  -u A )  e.  RR )
2723, 26resubcld 9227 . . . . 5  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  RR )
2827recnd 8877 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
2923, 26readdcld 8878 . . . . 5  |-  ( A  e.  RR  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  RR )
3029recnd 8877 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  CC )
31 2cn 9832 . . . . 5  |-  2  e.  CC
3231a1i 10 . . . 4  |-  ( A  e.  RR  ->  2  e.  CC )
3320, 6eqnetrrd 2479 . . . . 5  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  +  ( exp `  -u A ) )  /  2 )  =/=  0 )
34 2ne0 9845 . . . . . . 7  |-  2  =/=  0
3534a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  2  =/=  0 )
3630, 32, 35divne0bd 9564 . . . . 5  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  +  ( exp `  -u A ) )  =/=  0  <->  ( (
( exp `  A
)  +  ( exp `  -u A ) )  /  2 )  =/=  0 ) )
3733, 36mpbird 223 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  =/=  0 )
3828, 30, 32, 37, 35divcan7d 9580 . . 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
) ) ) )
3922, 38eqtrd 2328 . 2  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  (
( exp `  A
)  +  ( exp `  -u A ) ) ) )
4024rpefcld 12401 . . . . . 6  |-  ( A  e.  RR  ->  ( exp `  -u A )  e.  RR+ )
4123, 40ltsubrpd 10434 . . . . 5  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  <  ( exp `  A
) )
4223, 40ltaddrpd 10435 . . . . 5  |-  ( A  e.  RR  ->  ( exp `  A )  < 
( ( exp `  A
)  +  ( exp `  -u A ) ) )
4327, 23, 29, 41, 42lttrd 8993 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  <  ( ( exp `  A )  +  ( exp `  -u A
) ) )
4430mulid1d 8868 . . . 4  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  +  ( exp `  -u A ) )  x.  1 )  =  ( ( exp `  A
)  +  ( exp `  -u A ) ) )
4543, 44breqtrrd 4065 . . 3  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  <  ( ( ( exp `  A )  +  ( exp `  -u A
) )  x.  1 ) )
46 1re 8853 . . . . 5  |-  1  e.  RR
4746a1i 10 . . . 4  |-  ( A  e.  RR  ->  1  e.  RR )
48 efgt0 12399 . . . . 5  |-  ( A  e.  RR  ->  0  <  ( exp `  A
) )
49 efgt0 12399 . . . . . 6  |-  ( -u A  e.  RR  ->  0  <  ( exp `  -u A
) )
5024, 49syl 15 . . . . 5  |-  ( A  e.  RR  ->  0  <  ( exp `  -u A
) )
5123, 26, 48, 50addgt0d 9363 . . . 4  |-  ( A  e.  RR  ->  0  <  ( ( exp `  A
)  +  ( exp `  -u A ) ) )
52 ltdivmul 9644 . . . 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 ) ) )
5327, 47, 29, 51, 52syl112anc 1186 . . 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 ) ) )
5445, 53mpbird 223 . 2  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  ( ( exp `  A )  +  ( exp `  -u A
) ) )  <  1 )
5539, 54eqbrtrd 4059 1  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  <  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   expce 12359   sincsin 12361   cosccos 12362   tanctan 12363
This theorem is referenced by:  tanhbnd  12457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-tan 12369
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