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Theorem tanatan 20215
Description: The arctangent function is an inverse to  tan. (Contributed by Mario Carneiro, 2-Apr-2015.)
Assertion
Ref Expression
tanatan  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  A )

Proof of Theorem tanatan
StepHypRef Expression
1 atancl 20177 . . 3  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
2 2efiatan 20214 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 ) )
32oveq1d 5873 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  +  1 ) )
4 2cn 9816 . . . . . . . . 9  |-  2  e.  CC
5 ax-icn 8796 . . . . . . . . 9  |-  _i  e.  CC
64, 5mulcli 8842 . . . . . . . 8  |-  ( 2  x.  _i )  e.  CC
76a1i 10 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  e.  CC )
8 atandm 20172 . . . . . . . . 9  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
98simp1bi 970 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  A  e.  CC )
10 addcl 8819 . . . . . . . 8  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  +  _i )  e.  CC )
119, 5, 10sylancl 643 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  e.  CC )
12 subneg 9096 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  -  -u _i )  =  ( A  +  _i ) )
139, 5, 12sylancl 643 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =  ( A  +  _i ) )
148simp2bi 971 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  A  =/=  -u _i )
155negcli 9114 . . . . . . . . . 10  |-  -u _i  e.  CC
16 subeq0 9073 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =  0  <->  A  =  -u _i ) )
1716necon3bid 2481 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
189, 15, 17sylancl 643 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
1914, 18mpbird 223 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =/=  0 )
2013, 19eqnetrrd 2466 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  =/=  0 )
217, 11, 20divcld 9536 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  /  ( A  +  _i ) )  e.  CC )
22 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
23 npcan 9060 . . . . . 6  |-  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 )  +  1 )  =  ( ( 2  x.  _i )  / 
( A  +  _i ) ) )
2421, 22, 23sylancl 643 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  +  1 )  =  ( ( 2  x.  _i )  /  ( A  +  _i )
) )
253, 24eqtrd 2315 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =  ( ( 2  x.  _i )  /  ( A  +  _i ) ) )
26 2ne0 9829 . . . . . . 7  |-  2  =/=  0
27 ine0 9215 . . . . . . 7  |-  _i  =/=  0
284, 5, 26, 27mulne0i 9411 . . . . . 6  |-  ( 2  x.  _i )  =/=  0
2928a1i 10 . . . . 5  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  =/=  0 )
307, 11, 29, 20divne0d 9552 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  /  ( A  +  _i ) )  =/=  0
)
3125, 30eqnetrd 2464 . . 3  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =/=  0 )
32 tanval3 12414 . . 3  |-  ( ( (arctan `  A )  e.  CC  /\  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =/=  0 )  ->  ( tan `  (arctan `  A ) )  =  ( ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) ) )
331, 31, 32syl2anc 642 . 2  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) ) )
342oveq1d 5873 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 ) )
3522a1i 10 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  1  e.  CC )
3621, 35, 35subsub4d 9188 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  (
1  +  1 ) ) )
37 df-2 9804 . . . . . . . 8  |-  2  =  ( 1  +  1 )
3837oveq2i 5869 . . . . . . 7  |-  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
1  +  1 ) )
3936, 38syl6eqr 2333 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  2 ) )
4034, 39eqtrd 2315 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
41 mulcl 8821 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  ( A  +  _i )  e.  CC )  ->  ( 2  x.  ( A  +  _i )
)  e.  CC )
424, 11, 41sylancr 644 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( A  +  _i ) )  e.  CC )
437, 42, 11, 20divsubdird 9575 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  / 
( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  ( ( 2  x.  ( A  +  _i ) )  /  ( A  +  _i ) ) ) )
44 mulneg12 9218 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( -u 2  x.  A )  =  ( 2  x.  -u A
) )
454, 9, 44sylancr 644 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( -u
2  x.  A )  =  ( 2  x.  -u A ) )
46 negsub 9095 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  -u A )  =  ( _i  -  A ) )
475, 9, 46sylancr 644 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( _i  +  -u A )  =  ( _i  -  A
) )
4847oveq1d 5873 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  +  -u A
)  -  _i )  =  ( ( _i 
-  A )  -  _i ) )
499negcld 9144 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  -u A  e.  CC )
50 pncan2 9058 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  -u A  e.  CC )  ->  ( ( _i  +  -u A )  -  _i )  =  -u A
)
515, 49, 50sylancr 644 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  +  -u A
)  -  _i )  =  -u A )
525a1i 10 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
5352, 9, 52subsub4d 9188 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  -  A )  -  _i )  =  ( _i  -  ( A  +  _i )
) )
5448, 51, 533eqtr3rd 2324 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( _i 
-  ( A  +  _i ) )  =  -u A )
5554oveq2d 5874 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  -  ( A  +  _i ) ) )  =  ( 2  x.  -u A
) )
564a1i 10 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  2  e.  CC )
5756, 52, 11subdid 9235 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  -  ( A  +  _i ) ) )  =  ( ( 2  x.  _i )  -  (
2  x.  ( A  +  _i ) ) ) )
5845, 55, 573eqtr2rd 2322 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  =  ( -u 2  x.  A ) )
5958oveq1d 5873 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  / 
( A  +  _i ) )  =  ( ( -u 2  x.  A )  /  ( A  +  _i )
) )
6056, 11, 20divcan4d 9542 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( A  +  _i ) )  /  ( A  +  _i ) )  =  2 )
6160oveq2d 5874 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
( 2  x.  ( A  +  _i )
)  /  ( A  +  _i ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
6243, 59, 613eqtr3d 2323 . . . . 5  |-  ( A  e.  dom arctan  ->  ( (
-u 2  x.  A
)  /  ( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
6340, 62eqtr4d 2318 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( (
-u 2  x.  A
)  /  ( A  +  _i ) ) )
6425oveq2d 5874 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  +  1 ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i )
) ) )
655, 4, 5mul12i 9007 . . . . . . . 8  |-  ( _i  x.  ( 2  x.  _i ) )  =  ( 2  x.  (
_i  x.  _i )
)
66 ixi 9397 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
6766oveq2i 5869 . . . . . . . 8  |-  ( 2  x.  ( _i  x.  _i ) )  =  ( 2  x.  -u 1
)
6822negcli 9114 . . . . . . . . 9  |-  -u 1  e.  CC
694mulm1i 9224 . . . . . . . . 9  |-  ( -u
1  x.  2 )  =  -u 2
7068, 4, 69mulcomli 8844 . . . . . . . 8  |-  ( 2  x.  -u 1 )  = 
-u 2
7165, 67, 703eqtri 2307 . . . . . . 7  |-  ( _i  x.  ( 2  x.  _i ) )  = 
-u 2
7271oveq1i 5868 . . . . . 6  |-  ( ( _i  x.  ( 2  x.  _i ) )  /  ( A  +  _i ) )  =  (
-u 2  /  ( A  +  _i )
)
7352, 7, 11, 20divassd 9571 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  ( 2  x.  _i ) )  /  ( A  +  _i ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i ) ) ) )
7472, 73syl5eqr 2329 . . . . 5  |-  ( A  e.  dom arctan  ->  ( -u
2  /  ( A  +  _i ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i )
) ) )
7564, 74eqtr4d 2318 . . . 4  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  +  1 ) )  =  ( -u 2  /  ( A  +  _i ) ) )
7663, 75oveq12d 5876 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) )  =  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) ) )
774negcli 9114 . . . . . 6  |-  -u 2  e.  CC
78 mulcl 8821 . . . . . 6  |-  ( (
-u 2  e.  CC  /\  A  e.  CC )  ->  ( -u 2  x.  A )  e.  CC )
7977, 9, 78sylancr 644 . . . . 5  |-  ( A  e.  dom arctan  ->  ( -u
2  x.  A )  e.  CC )
8077a1i 10 . . . . 5  |-  ( A  e.  dom arctan  ->  -u 2  e.  CC )
814, 26negne0i 9121 . . . . . 6  |-  -u 2  =/=  0
8281a1i 10 . . . . 5  |-  ( A  e.  dom arctan  ->  -u 2  =/=  0 )
8379, 80, 11, 82, 20divcan7d 9564 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) )  =  ( (
-u 2  x.  A
)  /  -u 2
) )
849, 80, 82divcan3d 9541 . . . 4  |-  ( A  e.  dom arctan  ->  ( (
-u 2  x.  A
)  /  -u 2
)  =  A )
8583, 84eqtrd 2315 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) )  =  A )
8676, 85eqtrd 2315 . 2  |-  ( A  e.  dom arctan  ->  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) )  =  A )
8733, 86eqtrd 2315 1  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   dom cdm 4689   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   expce 12343   tanctan 12347  arctancatan 20160
This theorem is referenced by:  atantanb  20220  atanord  20223
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-iin 3908  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-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  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-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  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  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-atan 20163
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