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Mirrors > Home > MPE Home > Th. List > Mathboxes > sinhval-named | Structured version Visualization version GIF version |
Description: Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 44831. See sinhval 15506 for a theorem to convert this further. See sinh-conventional 44837 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.) |
Ref | Expression |
---|---|
sinhval-named | ⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7163 | . . . 4 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
2 | 1 | fveq2d 6673 | . . 3 ⊢ (𝑥 = 𝐴 → (sin‘(i · 𝑥)) = (sin‘(i · 𝐴))) |
3 | 2 | oveq1d 7170 | . 2 ⊢ (𝑥 = 𝐴 → ((sin‘(i · 𝑥)) / i) = ((sin‘(i · 𝐴)) / i)) |
4 | df-sinh 44831 | . 2 ⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i)) | |
5 | ovex 7188 | . 2 ⊢ ((sin‘(i · 𝐴)) / i) ∈ V | |
6 | 3, 4, 5 | fvmpt 6767 | 1 ⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ici 10538 · cmul 10541 / cdiv 11296 sincsin 15416 sinhcsinh 44828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-sinh 44831 |
This theorem is referenced by: sinh-conventional 44837 sinhpcosh 44838 |
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