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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 47941. See coshval 16105 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 7420 | . . 3 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
2 | 1 | fveq2d 6895 | . 2 ⊢ (𝑥 = 𝐴 → (cos‘(i · 𝑥)) = (cos‘(i · 𝐴))) |
3 | df-cosh 47941 | . 2 ⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i · 𝑥))) | |
4 | fvex 6904 | . 2 ⊢ (cos‘(i · 𝐴)) ∈ V | |
5 | 2, 3, 4 | fvmpt 6998 | 1 ⊢ (𝐴 ∈ ℂ → (cosh‘𝐴) = (cos‘(i · 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ici 11118 · cmul 11121 cosccos 16015 coshccosh 47938 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-cosh 47941 |
This theorem is referenced by: sinhpcosh 47947 |
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