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Mirrors > Home > MPE Home > Th. List > retanhcl | Structured version Visualization version GIF version |
Description: The hyperbolic tangent of a real number is real. (Contributed by Mario Carneiro, 4-Apr-2015.) |
Ref | Expression |
---|---|
retanhcl | ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 11151 | . . . . . 6 ⊢ i ∈ ℂ | |
2 | recn 11182 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | mulcl 11176 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
4 | 1, 2, 3 | sylancr 587 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
5 | rpcoshcl 16082 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ+) | |
6 | 5 | rpne0d 13003 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ≠ 0) |
7 | tanval 16053 | . . . . 5 ⊢ (((i · 𝐴) ∈ ℂ ∧ (cos‘(i · 𝐴)) ≠ 0) → (tan‘(i · 𝐴)) = ((sin‘(i · 𝐴)) / (cos‘(i · 𝐴)))) | |
8 | 4, 6, 7 | syl2anc 584 | . . . 4 ⊢ (𝐴 ∈ ℝ → (tan‘(i · 𝐴)) = ((sin‘(i · 𝐴)) / (cos‘(i · 𝐴)))) |
9 | 8 | oveq1d 7408 | . . 3 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) = (((sin‘(i · 𝐴)) / (cos‘(i · 𝐴))) / i)) |
10 | 4 | sincld 16055 | . . . 4 ⊢ (𝐴 ∈ ℝ → (sin‘(i · 𝐴)) ∈ ℂ) |
11 | recoshcl 16083 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ) | |
12 | 11 | recnd 11224 | . . . 4 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℂ) |
13 | 1 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℝ → i ∈ ℂ) |
14 | ine0 11631 | . . . . 5 ⊢ i ≠ 0 | |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℝ → i ≠ 0) |
16 | 10, 12, 13, 6, 15 | divdiv32d 11997 | . . 3 ⊢ (𝐴 ∈ ℝ → (((sin‘(i · 𝐴)) / (cos‘(i · 𝐴))) / i) = (((sin‘(i · 𝐴)) / i) / (cos‘(i · 𝐴)))) |
17 | 9, 16 | eqtrd 2771 | . 2 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) = (((sin‘(i · 𝐴)) / i) / (cos‘(i · 𝐴)))) |
18 | resinhcl 16081 | . . 3 ⊢ (𝐴 ∈ ℝ → ((sin‘(i · 𝐴)) / i) ∈ ℝ) | |
19 | 18, 5 | rerpdivcld 13029 | . 2 ⊢ (𝐴 ∈ ℝ → (((sin‘(i · 𝐴)) / i) / (cos‘(i · 𝐴))) ∈ ℝ) |
20 | 17, 19 | eqeltrd 2832 | 1 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ‘cfv 6532 (class class class)co 7393 ℂcc 11090 ℝcr 11091 0cc0 11092 ici 11094 · cmul 11097 / cdiv 11853 sincsin 15989 cosccos 15990 tanctan 15991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-inf2 9618 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-pm 8806 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-sup 9419 df-inf 9420 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-n0 12455 df-z 12541 df-uz 12805 df-rp 12957 df-ico 13312 df-fz 13467 df-fzo 13610 df-fl 13739 df-seq 13949 df-exp 14010 df-fac 14216 df-bc 14245 df-hash 14273 df-shft 14996 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-limsup 15397 df-clim 15414 df-rlim 15415 df-sum 15615 df-ef 15993 df-sin 15995 df-cos 15996 df-tan 15997 |
This theorem is referenced by: tanhbnd 16086 tanregt0 25977 |
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