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Mirrors > Home > MPE Home > Th. List > Mathboxes > tanhval-named | Structured version Visualization version GIF version |
Description: Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 42476. (Contributed by David A. Wheeler, 10-May-2015.) |
Ref | Expression |
---|---|
tanhval-named | ⊢ (𝐴 ∈ (◡cosh “ (ℂ ∖ {0})) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6658 | . . . 4 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
2 | 1 | fveq2d 6195 | . . 3 ⊢ (𝑥 = 𝐴 → (tan‘(i · 𝑥)) = (tan‘(i · 𝐴))) |
3 | 2 | oveq1d 6665 | . 2 ⊢ (𝑥 = 𝐴 → ((tan‘(i · 𝑥)) / i) = ((tan‘(i · 𝐴)) / i)) |
4 | df-tanh 42476 | . 2 ⊢ tanh = (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i)) | |
5 | ovex 6678 | . 2 ⊢ ((tan‘(i · 𝐴)) / i) ∈ V | |
6 | 3, 4, 5 | fvmpt 6282 | 1 ⊢ (𝐴 ∈ (◡cosh “ (ℂ ∖ {0})) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 {csn 4177 ◡ccnv 5113 “ cima 5117 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 ici 9938 · cmul 9941 / cdiv 10684 tanctan 14796 coshccosh 42472 tanhctanh 42473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-tanh 42476 |
This theorem is referenced by: (None) |
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