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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 15512 | . 2 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ ℝ) | |
2 | ax-icn 10596 | . . . . . . . 8 ⊢ i ∈ ℂ | |
3 | recn 10627 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | mulcl 10621 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 589 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
6 | rpcoshcl 15510 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ+) | |
7 | 6 | rpne0d 12437 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ≠ 0) |
8 | 5, 7 | tancld 15485 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (tan‘(i · 𝐴)) ∈ ℂ) |
9 | 2 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → i ∈ ℂ) |
10 | ine0 11075 | . . . . . . 7 ⊢ i ≠ 0 | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → i ≠ 0) |
12 | 8, 9, 11 | divnegd 11429 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) = (-(tan‘(i · 𝐴)) / i)) |
13 | mulneg2 11077 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴)) | |
14 | 2, 3, 13 | sylancr 589 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (i · -𝐴) = -(i · 𝐴)) |
15 | 14 | fveq2d 6674 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (tan‘(i · -𝐴)) = (tan‘-(i · 𝐴))) |
16 | tanneg 15501 | . . . . . . . 8 ⊢ (((i · 𝐴) ∈ ℂ ∧ (cos‘(i · 𝐴)) ≠ 0) → (tan‘-(i · 𝐴)) = -(tan‘(i · 𝐴))) | |
17 | 5, 7, 16 | syl2anc 586 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (tan‘-(i · 𝐴)) = -(tan‘(i · 𝐴))) |
18 | 15, 17 | eqtrd 2856 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (tan‘(i · -𝐴)) = -(tan‘(i · 𝐴))) |
19 | 18 | oveq1d 7171 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · -𝐴)) / i) = (-(tan‘(i · 𝐴)) / i)) |
20 | 12, 19 | eqtr4d 2859 | . . . 4 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) = ((tan‘(i · -𝐴)) / i)) |
21 | renegcl 10949 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
22 | tanhlt1 15513 | . . . . 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 5088 | . . 3 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) < 1) |
25 | 1re 10641 | . . . 4 ⊢ 1 ∈ ℝ | |
26 | ltnegcon1 11141 | . . . 4 ⊢ ((((tan‘(i · 𝐴)) / i) ∈ ℝ ∧ 1 ∈ ℝ) → (-((tan‘(i · 𝐴)) / i) < 1 ↔ -1 < ((tan‘(i · 𝐴)) / i))) | |
27 | 1, 25, 26 | sylancl 588 | . . 3 ⊢ (𝐴 ∈ ℝ → (-((tan‘(i · 𝐴)) / i) < 1 ↔ -1 < ((tan‘(i · 𝐴)) / i))) |
28 | 24, 27 | mpbid 234 | . 2 ⊢ (𝐴 ∈ ℝ → -1 < ((tan‘(i · 𝐴)) / i)) |
29 | tanhlt1 15513 | . 2 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) < 1) | |
30 | neg1rr 11753 | . . . 4 ⊢ -1 ∈ ℝ | |
31 | 30 | rexri 10699 | . . 3 ⊢ -1 ∈ ℝ* |
32 | 25 | rexri 10699 | . . 3 ⊢ 1 ∈ ℝ* |
33 | elioo2 12780 | . . 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 690 | . 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 1339 | 1 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ (-1(,)1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 ici 10539 · cmul 10542 ℝ*cxr 10674 < clt 10675 -cneg 10871 / cdiv 11297 (,)cioo 12739 cosccos 15418 tanctan 15419 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-ioo 12743 df-ico 12745 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-tan 15425 |
This theorem is referenced by: tanregt0 25123 atantan 25501 |
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