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Mirrors > Home > MPE Home > Th. List > tanhbnd | Structured version Visualization version GIF version |
Description: The hyperbolic tangent of a real number is bounded by 1. (Contributed by Mario Carneiro, 4-Apr-2015.) |
Ref | Expression |
---|---|
tanhbnd | ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ (-1(,)1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retanhcl 16109 | . 2 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ ℝ) | |
2 | ax-icn 11175 | . . . . . . . 8 ⊢ i ∈ ℂ | |
3 | recn 11206 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | mulcl 11200 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 586 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
6 | rpcoshcl 16107 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ+) | |
7 | 6 | rpne0d 13028 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ≠ 0) |
8 | 5, 7 | tancld 16082 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (tan‘(i · 𝐴)) ∈ ℂ) |
9 | 2 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → i ∈ ℂ) |
10 | ine0 11656 | . . . . . . 7 ⊢ i ≠ 0 | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → i ≠ 0) |
12 | 8, 9, 11 | divnegd 12010 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) = (-(tan‘(i · 𝐴)) / i)) |
13 | mulneg2 11658 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴)) | |
14 | 2, 3, 13 | sylancr 586 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (i · -𝐴) = -(i · 𝐴)) |
15 | 14 | fveq2d 6895 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (tan‘(i · -𝐴)) = (tan‘-(i · 𝐴))) |
16 | tanneg 16098 | . . . . . . . 8 ⊢ (((i · 𝐴) ∈ ℂ ∧ (cos‘(i · 𝐴)) ≠ 0) → (tan‘-(i · 𝐴)) = -(tan‘(i · 𝐴))) | |
17 | 5, 7, 16 | syl2anc 583 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (tan‘-(i · 𝐴)) = -(tan‘(i · 𝐴))) |
18 | 15, 17 | eqtrd 2771 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (tan‘(i · -𝐴)) = -(tan‘(i · 𝐴))) |
19 | 18 | oveq1d 7427 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · -𝐴)) / i) = (-(tan‘(i · 𝐴)) / i)) |
20 | 12, 19 | eqtr4d 2774 | . . . 4 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) = ((tan‘(i · -𝐴)) / i)) |
21 | renegcl 11530 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
22 | tanhlt1 16110 | . . . . 5 ⊢ (-𝐴 ∈ ℝ → ((tan‘(i · -𝐴)) / i) < 1) | |
23 | 21, 22 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · -𝐴)) / i) < 1) |
24 | 20, 23 | eqbrtrd 5170 | . . 3 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) < 1) |
25 | 1re 11221 | . . . 4 ⊢ 1 ∈ ℝ | |
26 | ltnegcon1 11722 | . . . 4 ⊢ ((((tan‘(i · 𝐴)) / i) ∈ ℝ ∧ 1 ∈ ℝ) → (-((tan‘(i · 𝐴)) / i) < 1 ↔ -1 < ((tan‘(i · 𝐴)) / i))) | |
27 | 1, 25, 26 | sylancl 585 | . . 3 ⊢ (𝐴 ∈ ℝ → (-((tan‘(i · 𝐴)) / i) < 1 ↔ -1 < ((tan‘(i · 𝐴)) / i))) |
28 | 24, 27 | mpbid 231 | . 2 ⊢ (𝐴 ∈ ℝ → -1 < ((tan‘(i · 𝐴)) / i)) |
29 | tanhlt1 16110 | . 2 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) < 1) | |
30 | neg1rr 12334 | . . . 4 ⊢ -1 ∈ ℝ | |
31 | 30 | rexri 11279 | . . 3 ⊢ -1 ∈ ℝ* |
32 | 25 | rexri 11279 | . . 3 ⊢ 1 ∈ ℝ* |
33 | elioo2 13372 | . . 3 ⊢ ((-1 ∈ ℝ* ∧ 1 ∈ ℝ*) → (((tan‘(i · 𝐴)) / i) ∈ (-1(,)1) ↔ (((tan‘(i · 𝐴)) / i) ∈ ℝ ∧ -1 < ((tan‘(i · 𝐴)) / i) ∧ ((tan‘(i · 𝐴)) / i) < 1))) | |
34 | 31, 32, 33 | mp2an 689 | . 2 ⊢ (((tan‘(i · 𝐴)) / i) ∈ (-1(,)1) ↔ (((tan‘(i · 𝐴)) / i) ∈ ℝ ∧ -1 < ((tan‘(i · 𝐴)) / i) ∧ ((tan‘(i · 𝐴)) / i) < 1)) |
35 | 1, 28, 29, 34 | syl3anbrc 1342 | 1 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ (-1(,)1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ℝcr 11115 0cc0 11116 1c1 11117 ici 11118 · cmul 11121 ℝ*cxr 11254 < clt 11255 -cneg 11452 / cdiv 11878 (,)cioo 13331 cosccos 16015 tanctan 16016 |
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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 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-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-ioo 13335 df-ico 13337 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-fac 14241 df-bc 14270 df-hash 14298 df-shft 15021 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15640 df-ef 16018 df-sin 16020 df-cos 16021 df-tan 16022 |
This theorem is referenced by: tanregt0 26388 atantan 26769 |
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