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Theorem atandmtan 23451
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 13949 . 2  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  e.  CC )
2 tanval 13948 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
32oveq1d 6285 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( tan `  A
) ^ 2 )  =  ( ( ( sin `  A )  /  ( cos `  A
) ) ^ 2 ) )
4 sincl 13946 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
54adantr 463 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sin `  A
)  e.  CC )
6 coscl 13947 . . . . . 6  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
76adantr 463 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( cos `  A
)  e.  CC )
8 simpr 459 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( cos `  A
)  =/=  0 )
95, 7, 8sqdivd 12308 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A )  /  ( cos `  A ) ) ^ 2 )  =  ( ( ( sin `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
103, 9eqtrd 2495 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( tan `  A
) ^ 2 )  =  ( ( ( sin `  A ) ^ 2 )  / 
( ( cos `  A
) ^ 2 ) ) )
115sqcld 12293 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( sin `  A
) ^ 2 )  e.  CC )
127sqcld 12293 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( cos `  A
) ^ 2 )  e.  CC )
1312negcld 9909 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  -u ( ( cos `  A
) ^ 2 )  e.  CC )
1411, 12subnegd 9929 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A ) ^ 2 )  -  -u (
( cos `  A
) ^ 2 ) )  =  ( ( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) ) )
15 sincossq 13996 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
1615adantr 463 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A ) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
1714, 16eqtrd 2495 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A ) ^ 2 )  -  -u (
( cos `  A
) ^ 2 ) )  =  1 )
18 ax-1ne0 9550 . . . . . . . 8  |-  1  =/=  0
1918a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
1  =/=  0 )
2017, 19eqnetrd 2747 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A ) ^ 2 )  -  -u (
( cos `  A
) ^ 2 ) )  =/=  0 )
2111, 13, 20subne0ad 9933 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( sin `  A
) ^ 2 )  =/=  -u ( ( cos `  A ) ^ 2 ) )
2212mulm1d 10004 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( -u 1  x.  (
( cos `  A
) ^ 2 ) )  =  -u (
( cos `  A
) ^ 2 ) )
2321, 22neeqtrrd 2754 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( sin `  A
) ^ 2 )  =/=  ( -u 1  x.  ( ( cos `  A
) ^ 2 ) ) )
24 neg1cn 10635 . . . . . . 7  |-  -u 1  e.  CC
2524a1i 11 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  -u 1  e.  CC )
26 sqne0 12219 . . . . . . . 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 483 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( cos `  A
) ^ 2 )  =/=  0 )
2911, 25, 12, 28divmul3d 10350 . . . . 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 2712 . . . 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 2747 . 2  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( tan `  A
) ^ 2 )  =/=  -u 1 )
33 atandm3 23409 . 2  |-  ( ( tan `  A )  e.  dom arctan  <->  ( ( tan `  A )  e.  CC  /\  ( ( tan `  A
) ^ 2 )  =/=  -u 1 ) )
341, 32, 33sylanbrc 662 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   dom cdm 4988   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796   -ucneg 9797    / cdiv 10202   2c2 10581   ^cexp 12151   sincsin 13884   cosccos 13885   tanctan 13886  arctancatan 23395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-ico 11538  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12093  df-exp 12152  df-fac 12339  df-bc 12366  df-hash 12391  df-shft 12985  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-limsup 13379  df-clim 13396  df-rlim 13397  df-sum 13594  df-ef 13888  df-sin 13890  df-cos 13891  df-tan 13892  df-atan 23398
This theorem is referenced by:  atantan  23454
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