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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 15504 | . 2 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ ℝ) | |
2 | ax-icn 10585 | . . . . . . . 8 ⊢ i ∈ ℂ | |
3 | recn 10616 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | mulcl 10610 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 590 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
6 | rpcoshcl 15502 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ+) | |
7 | 6 | rpne0d 12424 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ≠ 0) |
8 | 5, 7 | tancld 15477 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (tan‘(i · 𝐴)) ∈ ℂ) |
9 | 2 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → i ∈ ℂ) |
10 | ine0 11064 | . . . . . . 7 ⊢ i ≠ 0 | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → i ≠ 0) |
12 | 8, 9, 11 | divnegd 11418 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) = (-(tan‘(i · 𝐴)) / i)) |
13 | mulneg2 11066 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴)) | |
14 | 2, 3, 13 | sylancr 590 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (i · -𝐴) = -(i · 𝐴)) |
15 | 14 | fveq2d 6649 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (tan‘(i · -𝐴)) = (tan‘-(i · 𝐴))) |
16 | tanneg 15493 | . . . . . . . 8 ⊢ (((i · 𝐴) ∈ ℂ ∧ (cos‘(i · 𝐴)) ≠ 0) → (tan‘-(i · 𝐴)) = -(tan‘(i · 𝐴))) | |
17 | 5, 7, 16 | syl2anc 587 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (tan‘-(i · 𝐴)) = -(tan‘(i · 𝐴))) |
18 | 15, 17 | eqtrd 2833 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (tan‘(i · -𝐴)) = -(tan‘(i · 𝐴))) |
19 | 18 | oveq1d 7150 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · -𝐴)) / i) = (-(tan‘(i · 𝐴)) / i)) |
20 | 12, 19 | eqtr4d 2836 | . . . 4 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) = ((tan‘(i · -𝐴)) / i)) |
21 | renegcl 10938 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
22 | tanhlt1 15505 | . . . . 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 5052 | . . 3 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) < 1) |
25 | 1re 10630 | . . . 4 ⊢ 1 ∈ ℝ | |
26 | ltnegcon1 11130 | . . . 4 ⊢ ((((tan‘(i · 𝐴)) / i) ∈ ℝ ∧ 1 ∈ ℝ) → (-((tan‘(i · 𝐴)) / i) < 1 ↔ -1 < ((tan‘(i · 𝐴)) / i))) | |
27 | 1, 25, 26 | sylancl 589 | . . 3 ⊢ (𝐴 ∈ ℝ → (-((tan‘(i · 𝐴)) / i) < 1 ↔ -1 < ((tan‘(i · 𝐴)) / i))) |
28 | 24, 27 | mpbid 235 | . 2 ⊢ (𝐴 ∈ ℝ → -1 < ((tan‘(i · 𝐴)) / i)) |
29 | tanhlt1 15505 | . 2 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) < 1) | |
30 | neg1rr 11740 | . . . 4 ⊢ -1 ∈ ℝ | |
31 | 30 | rexri 10688 | . . 3 ⊢ -1 ∈ ℝ* |
32 | 25 | rexri 10688 | . . 3 ⊢ 1 ∈ ℝ* |
33 | elioo2 12767 | . . 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 691 | . 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 1340 | 1 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ (-1(,)1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 ici 10528 · cmul 10531 ℝ*cxr 10663 < clt 10664 -cneg 10860 / cdiv 11286 (,)cioo 12726 cosccos 15410 tanctan 15411 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ioo 12730 df-ico 12732 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-tan 15417 |
This theorem is referenced by: tanregt0 25131 atantan 25509 |
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