ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tanval2ap GIF version

Theorem tanval2ap 11721
Description: Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
Assertion
Ref Expression
tanval2ap ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (i Β· ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))))

Proof of Theorem tanval2ap
StepHypRef Expression
1 tanvalap 11716 . . 3 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜π΄) = ((sinβ€˜π΄) / (cosβ€˜π΄)))
2 2cn 8990 . . . . . . 7 2 ∈ β„‚
3 ax-icn 7906 . . . . . . 7 i ∈ β„‚
42, 3mulcomi 7963 . . . . . 6 (2 Β· i) = (i Β· 2)
54oveq2i 5886 . . . . 5 (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (2 Β· i)) = (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (i Β· 2))
6 sinval 11710 . . . . . 6 (𝐴 ∈ β„‚ β†’ (sinβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (2 Β· i)))
76adantr 276 . . . . 5 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (sinβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (2 Β· i)))
8 simpl 109 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ 𝐴 ∈ β„‚)
9 mulcl 7938 . . . . . . . . 9 ((i ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (i Β· 𝐴) ∈ β„‚)
103, 8, 9sylancr 414 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (i Β· 𝐴) ∈ β„‚)
11 efcl 11672 . . . . . . . 8 ((i Β· 𝐴) ∈ β„‚ β†’ (expβ€˜(i Β· 𝐴)) ∈ β„‚)
1210, 11syl 14 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (expβ€˜(i Β· 𝐴)) ∈ β„‚)
13 negicn 8158 . . . . . . . . 9 -i ∈ β„‚
14 mulcl 7938 . . . . . . . . 9 ((-i ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (-i Β· 𝐴) ∈ β„‚)
1513, 8, 14sylancr 414 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (-i Β· 𝐴) ∈ β„‚)
16 efcl 11672 . . . . . . . 8 ((-i Β· 𝐴) ∈ β„‚ β†’ (expβ€˜(-i Β· 𝐴)) ∈ β„‚)
1715, 16syl 14 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (expβ€˜(-i Β· 𝐴)) ∈ β„‚)
1812, 17subcld 8268 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ ((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) ∈ β„‚)
193a1i 9 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ i ∈ β„‚)
202a1i 9 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ 2 ∈ β„‚)
21 iap0 9142 . . . . . . 7 i # 0
2221a1i 9 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ i # 0)
23 2ap0 9012 . . . . . . 7 2 # 0
2423a1i 9 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ 2 # 0)
2518, 19, 20, 22, 24divdivap1d 8779 . . . . 5 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ ((((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / i) / 2) = (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (i Β· 2)))
265, 7, 253eqtr4a 2236 . . . 4 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (sinβ€˜π΄) = ((((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / i) / 2))
27 cosval 11711 . . . . 5 (𝐴 ∈ β„‚ β†’ (cosβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) / 2))
2827adantr 276 . . . 4 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (cosβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) / 2))
2926, 28oveq12d 5893 . . 3 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ ((sinβ€˜π΄) / (cosβ€˜π΄)) = (((((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / i) / 2) / (((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) / 2)))
301, 29eqtrd 2210 . 2 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜π΄) = (((((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / i) / 2) / (((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) / 2)))
3118, 19, 22divclapd 8747 . . 3 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / i) ∈ β„‚)
3212, 17addcld 7977 . . 3 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) ∈ β„‚)
33 simpr 110 . . . . 5 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (cosβ€˜π΄) # 0)
3428, 33eqbrtrrd 4028 . . . 4 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) / 2) # 0)
3532, 20, 24divap0bd 8759 . . . 4 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) # 0 ↔ (((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) / 2) # 0))
3634, 35mpbird 167 . . 3 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) # 0)
3731, 32, 20, 36, 24divcanap7d 8776 . 2 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (((((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / i) / 2) / (((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) / 2)) = ((((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / i) / ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴)))))
3818, 19, 32, 22, 36divdivap1d 8779 . 2 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ ((((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / i) / ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴)))) = (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (i Β· ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))))
3930, 37, 383eqtrd 2214 1 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) # 0) β†’ (tanβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) βˆ’ (expβ€˜(-i Β· 𝐴))) / (i Β· ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   class class class wbr 4004  β€˜cfv 5217  (class class class)co 5875  β„‚cc 7809  0cc0 7811  ici 7813   + caddc 7814   Β· cmul 7816   βˆ’ cmin 8128  -cneg 8129   # cap 8538   / cdiv 8629  2c2 8970  expce 11650  sincsin 11652  cosccos 11653  tanctan 11654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-frec 6392  df-1o 6417  df-oadd 6421  df-er 6535  df-en 6741  df-dom 6742  df-fin 6743  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-ico 9894  df-fz 10009  df-fzo 10143  df-seqfrec 10446  df-exp 10520  df-fac 10706  df-ihash 10756  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-clim 11287  df-sumdc 11362  df-ef 11656  df-sin 11658  df-cos 11659  df-tan 11660
This theorem is referenced by:  tanval3ap  11722
  Copyright terms: Public domain W3C validator