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Theorem tanhbnd 12682
Description: The hyperbolic tangent of a real number is bounded by  1. (Contributed by Mario Carneiro, 4-Apr-2015.)
Assertion
Ref Expression
tanhbnd  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  e.  ( -u 1 (,) 1 ) )

Proof of Theorem tanhbnd
StepHypRef Expression
1 retanhcl 12680 . 2  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  e.  RR )
2 ax-icn 8975 . . . . . . . 8  |-  _i  e.  CC
3 recn 9006 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
4 mulcl 9000 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
52, 3, 4sylancr 645 . . . . . . 7  |-  ( A  e.  RR  ->  (
_i  x.  A )  e.  CC )
6 rpcoshcl 12678 . . . . . . . 8  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR+ )
76rpne0d 10578 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  =/=  0 )
85, 7tancld 12653 . . . . . 6  |-  ( A  e.  RR  ->  ( tan `  ( _i  x.  A ) )  e.  CC )
92a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  _i  e.  CC )
10 ine0 9394 . . . . . . 7  |-  _i  =/=  0
1110a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  _i  =/=  0 )
128, 9, 11divnegd 9728 . . . . 5  |-  ( A  e.  RR  ->  -u (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( -u ( tan `  ( _i  x.  A ) )  /  _i ) )
13 mulneg2 9396 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  -u A
)  =  -u (
_i  x.  A )
)
142, 3, 13sylancr 645 . . . . . . . 8  |-  ( A  e.  RR  ->  (
_i  x.  -u A )  =  -u ( _i  x.  A ) )
1514fveq2d 5665 . . . . . . 7  |-  ( A  e.  RR  ->  ( tan `  ( _i  x.  -u A ) )  =  ( tan `  -u (
_i  x.  A )
) )
16 tanneg 12669 . . . . . . . 8  |-  ( ( ( _i  x.  A
)  e.  CC  /\  ( cos `  ( _i  x.  A ) )  =/=  0 )  -> 
( tan `  -u (
_i  x.  A )
)  =  -u ( tan `  ( _i  x.  A ) ) )
175, 7, 16syl2anc 643 . . . . . . 7  |-  ( A  e.  RR  ->  ( tan `  -u ( _i  x.  A ) )  = 
-u ( tan `  (
_i  x.  A )
) )
1815, 17eqtrd 2412 . . . . . 6  |-  ( A  e.  RR  ->  ( tan `  ( _i  x.  -u A ) )  = 
-u ( tan `  (
_i  x.  A )
) )
1918oveq1d 6028 . . . . 5  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  -u A ) )  /  _i )  =  ( -u ( tan `  ( _i  x.  A ) )  /  _i ) )
2012, 19eqtr4d 2415 . . . 4  |-  ( A  e.  RR  ->  -u (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( tan `  ( _i  x.  -u A
) )  /  _i ) )
21 renegcl 9289 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
22 tanhlt1 12681 . . . . 5  |-  ( -u A  e.  RR  ->  ( ( tan `  (
_i  x.  -u A ) )  /  _i )  <  1 )
2321, 22syl 16 . . . 4  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  -u A ) )  /  _i )  <  1 )
2420, 23eqbrtrd 4166 . . 3  |-  ( A  e.  RR  ->  -u (
( tan `  (
_i  x.  A )
)  /  _i )  <  1 )
25 1re 9016 . . . 4  |-  1  e.  RR
26 ltnegcon1 9454 . . . 4  |-  ( ( ( ( tan `  (
_i  x.  A )
)  /  _i )  e.  RR  /\  1  e.  RR )  ->  ( -u ( ( tan `  (
_i  x.  A )
)  /  _i )  <  1  <->  -u 1  < 
( ( tan `  (
_i  x.  A )
)  /  _i ) ) )
271, 25, 26sylancl 644 . . 3  |-  ( A  e.  RR  ->  ( -u ( ( tan `  (
_i  x.  A )
)  /  _i )  <  1  <->  -u 1  < 
( ( tan `  (
_i  x.  A )
)  /  _i ) ) )
2824, 27mpbid 202 . 2  |-  ( A  e.  RR  ->  -u 1  <  ( ( tan `  (
_i  x.  A )
)  /  _i ) )
29 tanhlt1 12681 . 2  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  <  1 )
3025renegcli 9287 . . . 4  |-  -u 1  e.  RR
3130rexri 9063 . . 3  |-  -u 1  e.  RR*
3225rexri 9063 . . 3  |-  1  e.  RR*
33 elioo2 10882 . . 3  |-  ( (
-u 1  e.  RR*  /\  1  e.  RR* )  ->  ( ( ( tan `  ( _i  x.  A
) )  /  _i )  e.  ( -u 1 (,) 1 )  <->  ( (
( tan `  (
_i  x.  A )
)  /  _i )  e.  RR  /\  -u 1  <  ( ( tan `  (
_i  x.  A )
)  /  _i )  /\  ( ( tan `  ( _i  x.  A
) )  /  _i )  <  1 ) ) )
3431, 32, 33mp2an 654 . 2  |-  ( ( ( tan `  (
_i  x.  A )
)  /  _i )  e.  ( -u 1 (,) 1 )  <->  ( (
( tan `  (
_i  x.  A )
)  /  _i )  e.  RR  /\  -u 1  <  ( ( tan `  (
_i  x.  A )
)  /  _i )  /\  ( ( tan `  ( _i  x.  A
) )  /  _i )  <  1 ) )
351, 28, 29, 34syl3anbrc 1138 1  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  e.  ( -u 1 (,) 1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   CCcc 8914   RRcr 8915   0cc0 8916   1c1 8917   _ici 8918    x. cmul 8921   RR*cxr 9045    < clt 9046   -ucneg 9217    / cdiv 9602   (,)cioo 10841   cosccos 12587   tanctan 12588
This theorem is referenced by:  tanregt0  20301  atantan  20623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995  ax-mulf 8996
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-ioo 10845  df-ico 10847  df-fz 10969  df-fzo 11059  df-fl 11122  df-seq 11244  df-exp 11303  df-fac 11487  df-bc 11514  df-hash 11539  df-shft 11802  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-limsup 12185  df-clim 12202  df-rlim 12203  df-sum 12400  df-ef 12590  df-sin 12592  df-cos 12593  df-tan 12594
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