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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 44917. See sinhval 15492 for a theorem to convert this further. See sinh-conventional 44923 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 7150 | . . . 4 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
2 | 1 | fveq2d 6660 | . . 3 ⊢ (𝑥 = 𝐴 → (sin‘(i · 𝑥)) = (sin‘(i · 𝐴))) |
3 | 2 | oveq1d 7157 | . 2 ⊢ (𝑥 = 𝐴 → ((sin‘(i · 𝑥)) / i) = ((sin‘(i · 𝐴)) / i)) |
4 | df-sinh 44917 | . 2 ⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i)) | |
5 | ovex 7175 | . 2 ⊢ ((sin‘(i · 𝐴)) / i) ∈ V | |
6 | 3, 4, 5 | fvmpt 6754 | 1 ⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6341 (class class class)co 7142 ℂcc 10521 ici 10525 · cmul 10528 / cdiv 11283 sincsin 15402 sinhcsinh 44914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-iota 6300 df-fun 6343 df-fv 6349 df-ov 7145 df-sinh 44917 |
This theorem is referenced by: sinh-conventional 44923 sinhpcosh 44924 |
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