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Theorem atandmtan 22443
Description: The tangent function has range contained in the domain of the arctangent. (Contributed by Mario Carneiro, 31-Mar-2015.)
Assertion
Ref Expression
atandmtan  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  e.  dom arctan )

Proof of Theorem atandmtan
StepHypRef Expression
1 tancl 13526 . 2  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  e.  CC )
2 tanval 13525 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
32oveq1d 6210 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( tan `  A
) ^ 2 )  =  ( ( ( sin `  A )  /  ( cos `  A
) ) ^ 2 ) )
4 sincl 13523 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
54adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sin `  A
)  e.  CC )
6 coscl 13524 . . . . . 6  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
76adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( cos `  A
)  e.  CC )
8 simpr 461 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( cos `  A
)  =/=  0 )
95, 7, 8sqdivd 12133 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A )  /  ( cos `  A ) ) ^ 2 )  =  ( ( ( sin `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
103, 9eqtrd 2493 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( tan `  A
) ^ 2 )  =  ( ( ( sin `  A ) ^ 2 )  / 
( ( cos `  A
) ^ 2 ) ) )
115sqcld 12118 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( sin `  A
) ^ 2 )  e.  CC )
127sqcld 12118 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( cos `  A
) ^ 2 )  e.  CC )
1312negcld 9812 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  -u ( ( cos `  A
) ^ 2 )  e.  CC )
1411, 12subnegd 9832 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A ) ^ 2 )  -  -u (
( cos `  A
) ^ 2 ) )  =  ( ( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) ) )
15 sincossq 13573 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
1615adantr 465 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A ) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
1714, 16eqtrd 2493 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A ) ^ 2 )  -  -u (
( cos `  A
) ^ 2 ) )  =  1 )
18 ax-1ne0 9457 . . . . . . . 8  |-  1  =/=  0
1918a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
1  =/=  0 )
2017, 19eqnetrd 2742 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A ) ^ 2 )  -  -u (
( cos `  A
) ^ 2 ) )  =/=  0 )
2111, 13, 20subne0ad 9836 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( sin `  A
) ^ 2 )  =/=  -u ( ( cos `  A ) ^ 2 ) )
2212mulm1d 9902 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( -u 1  x.  (
( cos `  A
) ^ 2 ) )  =  -u (
( cos `  A
) ^ 2 ) )
2321, 22neeqtrrd 2749 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( sin `  A
) ^ 2 )  =/=  ( -u 1  x.  ( ( cos `  A
) ^ 2 ) ) )
24 neg1cn 10531 . . . . . . 7  |-  -u 1  e.  CC
2524a1i 11 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  -u 1  e.  CC )
26 sqne0 12044 . . . . . . . 8  |-  ( ( cos `  A )  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  =/=  0  <->  ( cos `  A )  =/=  0
) )
276, 26syl 16 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  =/=  0  <->  ( cos `  A )  =/=  0
) )
2827biimpar 485 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( cos `  A
) ^ 2 )  =/=  0 )
2911, 25, 12, 28divmul3d 10247 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( ( sin `  A ) ^ 2 )  / 
( ( cos `  A
) ^ 2 ) )  =  -u 1  <->  ( ( sin `  A
) ^ 2 )  =  ( -u 1  x.  ( ( cos `  A
) ^ 2 ) ) ) )
3029necon3bid 2707 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( ( sin `  A ) ^ 2 )  / 
( ( cos `  A
) ^ 2 ) )  =/=  -u 1  <->  ( ( sin `  A
) ^ 2 )  =/=  ( -u 1  x.  ( ( cos `  A
) ^ 2 ) ) ) )
3123, 30mpbird 232 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  =/=  -u 1 )
3210, 31eqnetrd 2742 . 2  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( tan `  A
) ^ 2 )  =/=  -u 1 )
33 atandm3 22401 . 2  |-  ( ( tan `  A )  e.  dom arctan  <->  ( ( tan `  A )  e.  CC  /\  ( ( tan `  A
) ^ 2 )  =/=  -u 1 ) )
341, 32, 33sylanbrc 664 1  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  e.  dom arctan )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   dom cdm 4943   ` cfv 5521  (class class class)co 6195   CCcc 9386   0cc0 9388   1c1 9389    + caddc 9391    x. cmul 9393    - cmin 9701   -ucneg 9702    / cdiv 10099   2c2 10477   ^cexp 11977   sincsin 13462   cosccos 13463   tanctan 13464  arctancatan 22387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467  ax-mulf 9468
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-ico 11412  df-fz 11550  df-fzo 11661  df-fl 11754  df-seq 11919  df-exp 11978  df-fac 12164  df-bc 12191  df-hash 12216  df-shft 12669  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-limsup 13062  df-clim 13079  df-rlim 13080  df-sum 13277  df-ef 13466  df-sin 13468  df-cos 13469  df-tan 13470  df-atan 22390
This theorem is referenced by:  atantan  22446
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