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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 49253. See coshval 16191 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 7439 | . . 3 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
| 2 | 1 | fveq2d 6910 | . 2 ⊢ (𝑥 = 𝐴 → (cos‘(i · 𝑥)) = (cos‘(i · 𝐴))) | 
| 3 | df-cosh 49253 | . 2 ⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥))) | |
| 4 | fvex 6919 | . 2 ⊢ (cos‘(i · 𝐴)) ∈ V | |
| 5 | 2, 3, 4 | fvmpt 7016 | 1 ⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ici 11157 · cmul 11160 cosccos 16100 coshccosh 49250 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-cosh 49253 | 
| This theorem is referenced by: sinhpcosh 49259 | 
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