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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 44841. (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 7167 | . . . 4 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
2 | 1 | fveq2d 6677 | . . 3 ⊢ (𝑥 = 𝐴 → (tan‘(i · 𝑥)) = (tan‘(i · 𝐴))) |
3 | 2 | oveq1d 7174 | . 2 ⊢ (𝑥 = 𝐴 → ((tan‘(i · 𝑥)) / i) = ((tan‘(i · 𝐴)) / i)) |
4 | df-tanh 44841 | . 2 ⊢ tanh = (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i)) | |
5 | ovex 7192 | . 2 ⊢ ((tan‘(i · 𝐴)) / i) ∈ V | |
6 | 3, 4, 5 | fvmpt 6771 | 1 ⊢ (𝐴 ∈ (◡cosh “ (ℂ ∖ {0})) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∖ cdif 3936 {csn 4570 ◡ccnv 5557 “ cima 5561 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 0cc0 10540 ici 10542 · cmul 10545 / cdiv 11300 tanctan 15422 coshccosh 44837 tanhctanh 44838 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-tanh 44841 |
This theorem is referenced by: (None) |
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