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Theorem tanhlt1 13759
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 9552 . . . . . . 7  |-  _i  e.  CC
2 recn 9583 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
3 mulcl 9577 . . . . . . 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 13756 . . . . . . 7  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR+ )
65rpne0d 11262 . . . . . 6  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  =/=  0 )
7 tanval 13727 . . . . . 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 6300 . . . 4  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( ( sin `  ( _i  x.  A ) )  /  ( cos `  (
_i  x.  A )
) )  /  _i ) )
104sincld 13729 . . . . 5  |-  ( A  e.  RR  ->  ( sin `  ( _i  x.  A ) )  e.  CC )
11 recoshcl 13757 . . . . . 6  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  RR )
1211recnd 9623 . . . . 5  |-  ( A  e.  RR  ->  ( cos `  ( _i  x.  A ) )  e.  CC )
131a1i 11 . . . . 5  |-  ( A  e.  RR  ->  _i  e.  CC )
14 ine0 9993 . . . . . 6  |-  _i  =/=  0
1514a1i 11 . . . . 5  |-  ( A  e.  RR  ->  _i  =/=  0 )
1610, 12, 13, 6, 15divdiv32d 10346 . . . 4  |-  ( A  e.  RR  ->  (
( ( sin `  (
_i  x.  A )
)  /  ( cos `  ( _i  x.  A
) ) )  /  _i )  =  (
( ( sin `  (
_i  x.  A )
)  /  _i )  /  ( cos `  (
_i  x.  A )
) ) )
17 sinhval 13753 . . . . . 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 13754 . . . . . 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 6303 . . . 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 2512 . . 3  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( ( ( exp `  A
)  -  ( exp `  -u A ) )  /  2 )  / 
( ( ( exp `  A )  +  ( exp `  -u A
) )  /  2
) ) )
23 reefcl 13687 . . . . . 6  |-  ( A  e.  RR  ->  ( exp `  A )  e.  RR )
24 renegcl 9883 . . . . . . 7  |-  ( A  e.  RR  ->  -u A  e.  RR )
2524reefcld 13688 . . . . . 6  |-  ( A  e.  RR  ->  ( exp `  -u A )  e.  RR )
2623, 25resubcld 9988 . . . . 5  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  RR )
2726recnd 9623 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  e.  CC )
2823, 25readdcld 9624 . . . . 5  |-  ( A  e.  RR  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  RR )
2928recnd 9623 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  +  ( exp `  -u A ) )  e.  CC )
30 2cnd 10609 . . . 4  |-  ( A  e.  RR  ->  2  e.  CC )
3120, 6eqnetrrd 2761 . . . . 5  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  +  ( exp `  -u A ) )  /  2 )  =/=  0 )
32 2ne0 10629 . . . . . . 7  |-  2  =/=  0
3332a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  2  =/=  0 )
3429, 30, 33divne0bd 10333 . . . . 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 10349 . . 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 2508 . 2  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  =  ( ( ( exp `  A )  -  ( exp `  -u A
) )  /  (
( exp `  A
)  +  ( exp `  -u A ) ) ) )
3824rpefcld 13704 . . . . . 6  |-  ( A  e.  RR  ->  ( exp `  -u A )  e.  RR+ )
3923, 38ltsubrpd 11285 . . . . 5  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  <  ( exp `  A
) )
4023, 38ltaddrpd 11286 . . . . 5  |-  ( A  e.  RR  ->  ( exp `  A )  < 
( ( exp `  A
)  +  ( exp `  -u A ) ) )
4126, 23, 28, 39, 40lttrd 9743 . . . 4  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  <  ( ( exp `  A )  +  ( exp `  -u A
) ) )
4229mulid1d 9614 . . . 4  |-  ( A  e.  RR  ->  (
( ( exp `  A
)  +  ( exp `  -u A ) )  x.  1 )  =  ( ( exp `  A
)  +  ( exp `  -u A ) ) )
4341, 42breqtrrd 4473 . . 3  |-  ( A  e.  RR  ->  (
( exp `  A
)  -  ( exp `  -u A ) )  <  ( ( ( exp `  A )  +  ( exp `  -u A
) )  x.  1 ) )
44 1red 9612 . . . 4  |-  ( A  e.  RR  ->  1  e.  RR )
45 efgt0 13702 . . . . 5  |-  ( A  e.  RR  ->  0  <  ( exp `  A
) )
46 efgt0 13702 . . . . . 6  |-  ( -u A  e.  RR  ->  0  <  ( exp `  -u A
) )
4724, 46syl 16 . . . . 5  |-  ( A  e.  RR  ->  0  <  ( exp `  -u A
) )
4823, 25, 45, 47addgt0d 10128 . . . 4  |-  ( A  e.  RR  ->  0  <  ( ( exp `  A
)  +  ( exp `  -u A ) ) )
49 ltdivmul 10418 . . . 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 4467 1  |-  ( A  e.  RR  ->  (
( tan `  (
_i  x.  A )
)  /  _i )  <  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   CCcc 9491   RRcr 9492   0cc0 9493   1c1 9494   _ici 9495    + caddc 9496    x. cmul 9498    < clt 9629    - cmin 9806   -ucneg 9807    / cdiv 10207   2c2 10586   expce 13662   sincsin 13664   cosccos 13665   tanctan 13666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571  ax-addf 9572  ax-mulf 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-ico 11536  df-fz 11674  df-fzo 11794  df-fl 11898  df-seq 12077  df-exp 12136  df-fac 12323  df-bc 12350  df-hash 12375  df-shft 12866  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-limsup 13260  df-clim 13277  df-rlim 13278  df-sum 13475  df-ef 13668  df-sin 13670  df-cos 13671  df-tan 13672
This theorem is referenced by:  tanhbnd  13760
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