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Theorem sinval 16070
Description: Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)
Assertion
Ref Expression
sinval (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)))

Proof of Theorem sinval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7412 . . . . 5 (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
21fveq2d 6888 . . . 4 (𝑥 = 𝐴 → (exp‘(i · 𝑥)) = (exp‘(i · 𝐴)))
3 oveq2 7412 . . . . 5 (𝑥 = 𝐴 → (-i · 𝑥) = (-i · 𝐴))
43fveq2d 6888 . . . 4 (𝑥 = 𝐴 → (exp‘(-i · 𝑥)) = (exp‘(-i · 𝐴)))
52, 4oveq12d 7422 . . 3 (𝑥 = 𝐴 → ((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) = ((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))))
65oveq1d 7419 . 2 (𝑥 = 𝐴 → (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)))
7 df-sin 16017 . 2 sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
8 ovex 7437 . 2 (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)) ∈ V
96, 7, 8fvmpt 6991 1 (𝐴 ∈ ℂ → (sin‘𝐴) = (((exp‘(i · 𝐴)) − (exp‘(-i · 𝐴))) / (2 · i)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6536  (class class class)co 7404  cc 11107  ici 11111   · cmul 11114  cmin 11445  -cneg 11446   / cdiv 11872  2c2 12268  expce 16009  sincsin 16011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-sin 16017
This theorem is referenced by:  tanval2  16081  resinval  16083  sinneg  16094  efival  16100  sinhval  16102  sinadd  16112  dvsincos  25864  sinper  26367  sineq0  26409  efeq1  26413  sinasin  26772  sineq0ALT  44255
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