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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 47990. See sinhval 16096 for a theorem to convert this further. See sinh-conventional 47996 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 7410 | . . . 4 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
2 | 1 | fveq2d 6886 | . . 3 ⊢ (𝑥 = 𝐴 → (sin‘(i · 𝑥)) = (sin‘(i · 𝐴))) |
3 | 2 | oveq1d 7417 | . 2 ⊢ (𝑥 = 𝐴 → ((sin‘(i · 𝑥)) / i) = ((sin‘(i · 𝐴)) / i)) |
4 | df-sinh 47990 | . 2 ⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i · 𝑥)) / i)) | |
5 | ovex 7435 | . 2 ⊢ ((sin‘(i · 𝐴)) / i) ∈ V | |
6 | 3, 4, 5 | fvmpt 6989 | 1 ⊢ (𝐴 ∈ ℂ → (sinh‘𝐴) = ((sin‘(i · 𝐴)) / i)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6534 (class class class)co 7402 ℂcc 11105 ici 11109 · cmul 11112 / cdiv 11869 sincsin 16005 sinhcsinh 47987 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fv 6542 df-ov 7405 df-sinh 47990 |
This theorem is referenced by: sinh-conventional 47996 sinhpcosh 47997 |
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