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Mirrors > Home > MPE Home > Th. List > Mathboxes > coshval-named | Structured version Visualization version GIF version |
Description: Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 47265. See coshval 16042 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.) |
Ref | Expression |
---|---|
coshval-named | ⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7366 | . . 3 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
2 | 1 | fveq2d 6847 | . 2 ⊢ (𝑥 = 𝐴 → (cos‘(i · 𝑥)) = (cos‘(i · 𝐴))) |
3 | df-cosh 47265 | . 2 ⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥))) | |
4 | fvex 6856 | . 2 ⊢ (cos‘(i · 𝐴)) ∈ V | |
5 | 2, 3, 4 | fvmpt 6949 | 1 ⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6497 (class class class)co 7358 ℂcc 11054 ici 11058 · cmul 11061 cosccos 15952 coshccosh 47262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-cosh 47265 |
This theorem is referenced by: sinhpcosh 47271 |
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