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Theorem tanhbnd 12441
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 12439 . 2  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  e.  RR )
2 ax-icn 8796 . . . . . . . 8  |-  _i  e.  CC
3 recn 8827 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
4 mulcl 8821 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
52, 3, 4sylancr 644 . . . . . . 7  |-  ( A  e.  RR  ->  (
_i  x.  A )  e.  CC )
6 rpcoshcl 12437 . . . . . . . 8  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR+ )
76rpne0d 10395 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  =/=  0 )
85, 7tancld 12412 . . . . . 6  |-  ( A  e.  RR  ->  ( tan `  ( _i  x.  A ) )  e.  CC )
92a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  _i  e.  CC )
10 ine0 9215 . . . . . . 7  |-  _i  =/=  0
1110a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  _i  =/=  0 )
128, 9, 11divnegd 9549 . . . . 5  |-  ( A  e.  RR  ->  -u (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( -u ( tan `  ( _i  x.  A ) )  /  _i ) )
13 mulneg2 9217 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  -u A
)  =  -u (
_i  x.  A )
)
142, 3, 13sylancr 644 . . . . . . . 8  |-  ( A  e.  RR  ->  (
_i  x.  -u A )  =  -u ( _i  x.  A ) )
1514fveq2d 5529 . . . . . . 7  |-  ( A  e.  RR  ->  ( tan `  ( _i  x.  -u A ) )  =  ( tan `  -u (
_i  x.  A )
) )
16 tanneg 12428 . . . . . . . 8  |-  ( ( ( _i  x.  A
)  e.  CC  /\  ( cos `  ( _i  x.  A ) )  =/=  0 )  -> 
( tan `  -u (
_i  x.  A )
)  =  -u ( tan `  ( _i  x.  A ) ) )
175, 7, 16syl2anc 642 . . . . . . 7  |-  ( A  e.  RR  ->  ( tan `  -u ( _i  x.  A ) )  = 
-u ( tan `  (
_i  x.  A )
) )
1815, 17eqtrd 2315 . . . . . 6  |-  ( A  e.  RR  ->  ( tan `  ( _i  x.  -u A ) )  = 
-u ( tan `  (
_i  x.  A )
) )
1918oveq1d 5873 . . . . 5  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  -u A ) )  /  _i )  =  ( -u ( tan `  ( _i  x.  A ) )  /  _i ) )
2012, 19eqtr4d 2318 . . . 4  |-  ( A  e.  RR  ->  -u (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( tan `  ( _i  x.  -u A
) )  /  _i ) )
21 renegcl 9110 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
22 tanhlt1 12440 . . . . 5  |-  ( -u A  e.  RR  ->  ( ( tan `  (
_i  x.  -u A ) )  /  _i )  <  1 )
2321, 22syl 15 . . . 4  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  -u A ) )  /  _i )  <  1 )
2420, 23eqbrtrd 4043 . . 3  |-  ( A  e.  RR  ->  -u (
( tan `  (
_i  x.  A )
)  /  _i )  <  1 )
25 1re 8837 . . . 4  |-  1  e.  RR
26 ltnegcon1 9275 . . . 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 643 . . 3  |-  ( A  e.  RR  ->  ( -u ( ( tan `  (
_i  x.  A )
)  /  _i )  <  1  <->  -u 1  < 
( ( tan `  (
_i  x.  A )
)  /  _i ) ) )
2824, 27mpbid 201 . 2  |-  ( A  e.  RR  ->  -u 1  <  ( ( tan `  (
_i  x.  A )
)  /  _i ) )
29 tanhlt1 12440 . 2  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  <  1 )
30 ressxr 8876 . . . 4  |-  RR  C_  RR*
3125renegcli 9108 . . . 4  |-  -u 1  e.  RR
3230, 31sselii 3177 . . 3  |-  -u 1  e.  RR*
3330, 25sselii 3177 . . 3  |-  1  e.  RR*
34 elioo2 10697 . . 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 ) ) )
3532, 33, 34mp2an 653 . 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 ) )
361, 28, 29, 35syl3anbrc 1136 1  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  e.  ( -u 1 (,) 1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   _ici 8739    x. cmul 8742   RR*cxr 8866    < clt 8867   -ucneg 9038    / cdiv 9423   (,)cioo 10656   cosccos 12346   tanctan 12347
This theorem is referenced by:  tanregt0  19901  atantan  20219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ioo 10660  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-tan 12353
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