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Theorem tanhlt1 13907
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 9568 . . . . . . 7  |-  _i  e.  CC
2 recn 9599 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
3 mulcl 9593 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
41, 2, 3sylancr 663 . . . . . 6  |-  ( A  e.  RR  ->  (
_i  x.  A )  e.  CC )
5 rpcoshcl 13904 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR+ )
65rpne0d 11286 . . . . . 6  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  =/=  0 )
7 tanval 13875 . . . . . 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 661 . . . . 5  |-  ( A  e.  RR  ->  ( tan `  ( _i  x.  A ) )  =  ( ( sin `  (
_i  x.  A )
)  /  ( cos `  ( _i  x.  A
) ) ) )
98oveq1d 6311 . . . 4  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( ( sin `  ( _i  x.  A ) )  /  ( cos `  (
_i  x.  A )
) )  /  _i ) )
104sincld 13877 . . . . 5  |-  ( A  e.  RR  ->  ( sin `  ( _i  x.  A ) )  e.  CC )
11 recoshcl 13905 . . . . . 6  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR )
1211recnd 9639 . . . . 5  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  CC )
131a1i 11 . . . . 5  |-  ( A  e.  RR  ->  _i  e.  CC )
14 ine0 10013 . . . . . 6  |-  _i  =/=  0
1514a1i 11 . . . . 5  |-  ( A  e.  RR  ->  _i  =/=  0 )
1610, 12, 13, 6, 15divdiv32d 10366 . . . 4  |-  ( A  e.  RR  ->  (
( ( sin `  (
_i  x.  A )
)  /  ( cos `  ( _i  x.  A
) ) )  /  _i )  =  (
( ( sin `  (
_i  x.  A )
)  /  _i )  /  ( cos `  (
_i  x.  A )
) ) )
17 sinhval 13901 . . . . . 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 13902 . . . . . 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 6314 . . . 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 2502 . . 3  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  / 
( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
23 reefcl 13834 . . . . . 6  |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR )
24 renegcl 9901 . . . . . . 7  |-  ( A  e.  RR  ->  -u A  e.  RR )
2524reefcld 13835 . . . . . 6  |-  ( A  e.  RR  ->  ( exp `  -u A )  e.  RR )
2623, 25resubcld 10008 . . . . 5  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  RR )
2726recnd 9639 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
2823, 25readdcld 9640 . . . . 5  |-  ( A  e.  RR  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  RR )
2928recnd 9639 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  CC )
30 2cnd 10629 . . . 4  |-  ( A  e.  RR  ->  2  e.  CC )
3120, 6eqnetrrd 2751 . . . . 5  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  +  ( exp `  -u A ) )  /  2 )  =/=  0 )
32 2ne0 10649 . . . . . . 7  |-  2  =/=  0
3332a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  2  =/=  0 )
3429, 30, 33divne0bd 10353 . . . . 5  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  +  ( exp `  -u A ) )  =/=  0  <->  ( (
( exp `  A
)  +  ( exp `  -u A ) )  /  2 )  =/=  0 ) )
3531, 34mpbird 232 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  =/=  0 )
3627, 29, 30, 35, 33divcan7d 10369 . . 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
) ) ) )
3722, 36eqtrd 2498 . 2  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  (
( exp `  A
)  +  ( exp `  -u A ) ) ) )
3824rpefcld 13852 . . . . . 6  |-  ( A  e.  RR  ->  ( exp `  -u A )  e.  RR+ )
3923, 38ltsubrpd 11309 . . . . 5  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  <  ( exp `  A
) )
4023, 38ltaddrpd 11310 . . . . 5  |-  ( A  e.  RR  ->  ( exp `  A )  < 
( ( exp `  A
)  +  ( exp `  -u A ) ) )
4126, 23, 28, 39, 40lttrd 9760 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  <  ( ( exp `  A )  +  ( exp `  -u A
) ) )
4229mulid1d 9630 . . . 4  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  +  ( exp `  -u A ) )  x.  1 )  =  ( ( exp `  A
)  +  ( exp `  -u A ) ) )
4341, 42breqtrrd 4482 . . 3  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  <  ( ( ( exp `  A )  +  ( exp `  -u A
) )  x.  1 ) )
44 1red 9628 . . . 4  |-  ( A  e.  RR  ->  1  e.  RR )
45 efgt0 13850 . . . . 5  |-  ( A  e.  RR  ->  0  <  ( exp `  A
) )
46 efgt0 13850 . . . . . 6  |-  ( -u A  e.  RR  ->  0  <  ( exp `  -u A
) )
4724, 46syl 16 . . . . 5  |-  ( A  e.  RR  ->  0  <  ( exp `  -u A
) )
4823, 25, 45, 47addgt0d 10148 . . . 4  |-  ( A  e.  RR  ->  0  <  ( ( exp `  A
)  +  ( exp `  -u A ) ) )
49 ltdivmul 10438 . . . 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 ) ) )
5026, 44, 28, 48, 49syl112anc 1232 . . 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 ) ) )
5143, 50mpbird 232 . 2  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  -  ( exp `  -u A ) )  /  ( ( exp `  A )  +  ( exp `  -u A
) ) )  <  1 )
5237, 51eqbrtrd 4476 1  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  <  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510   _ici 9511    + caddc 9512    x. cmul 9514    < clt 9645    - cmin 9824   -ucneg 9825    / cdiv 10227   2c2 10606   expce 13809   sincsin 13811   cosccos 13812   tanctan 13813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-ico 11560  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815  df-sin 13817  df-cos 13818  df-tan 13819
This theorem is referenced by:  tanhbnd  13908
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