Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 14889 | . 2 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ ℝ) | |
2 | ax-icn 9995 | . . . . . . . 8 ⊢ i ∈ ℂ | |
3 | recn 10026 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | mulcl 10020 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 695 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
6 | rpcoshcl 14887 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ+) | |
7 | 6 | rpne0d 11877 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ≠ 0) |
8 | 5, 7 | tancld 14862 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (tan‘(i · 𝐴)) ∈ ℂ) |
9 | 2 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → i ∈ ℂ) |
10 | ine0 10465 | . . . . . . 7 ⊢ i ≠ 0 | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → i ≠ 0) |
12 | 8, 9, 11 | divnegd 10814 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) = (-(tan‘(i · 𝐴)) / i)) |
13 | mulneg2 10467 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴)) | |
14 | 2, 3, 13 | sylancr 695 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (i · -𝐴) = -(i · 𝐴)) |
15 | 14 | fveq2d 6195 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (tan‘(i · -𝐴)) = (tan‘-(i · 𝐴))) |
16 | tanneg 14878 | . . . . . . . 8 ⊢ (((i · 𝐴) ∈ ℂ ∧ (cos‘(i · 𝐴)) ≠ 0) → (tan‘-(i · 𝐴)) = -(tan‘(i · 𝐴))) | |
17 | 5, 7, 16 | syl2anc 693 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (tan‘-(i · 𝐴)) = -(tan‘(i · 𝐴))) |
18 | 15, 17 | eqtrd 2656 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (tan‘(i · -𝐴)) = -(tan‘(i · 𝐴))) |
19 | 18 | oveq1d 6665 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · -𝐴)) / i) = (-(tan‘(i · 𝐴)) / i)) |
20 | 12, 19 | eqtr4d 2659 | . . . 4 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) = ((tan‘(i · -𝐴)) / i)) |
21 | renegcl 10344 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
22 | tanhlt1 14890 | . . . . 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 4675 | . . 3 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) < 1) |
25 | 1re 10039 | . . . 4 ⊢ 1 ∈ ℝ | |
26 | ltnegcon1 10529 | . . . 4 ⊢ ((((tan‘(i · 𝐴)) / i) ∈ ℝ ∧ 1 ∈ ℝ) → (-((tan‘(i · 𝐴)) / i) < 1 ↔ -1 < ((tan‘(i · 𝐴)) / i))) | |
27 | 1, 25, 26 | sylancl 694 | . . 3 ⊢ (𝐴 ∈ ℝ → (-((tan‘(i · 𝐴)) / i) < 1 ↔ -1 < ((tan‘(i · 𝐴)) / i))) |
28 | 24, 27 | mpbid 222 | . 2 ⊢ (𝐴 ∈ ℝ → -1 < ((tan‘(i · 𝐴)) / i)) |
29 | tanhlt1 14890 | . 2 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) < 1) | |
30 | neg1rr 11125 | . . . 4 ⊢ -1 ∈ ℝ | |
31 | 30 | rexri 10097 | . . 3 ⊢ -1 ∈ ℝ* |
32 | 25 | rexri 10097 | . . 3 ⊢ 1 ∈ ℝ* |
33 | elioo2 12216 | . . 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 708 | . 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 1246 | 1 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ (-1(,)1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 ici 9938 · cmul 9941 ℝ*cxr 10073 < clt 10074 -cneg 10267 / cdiv 10684 (,)cioo 12175 cosccos 14795 tanctan 14796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ioo 12179 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-tan 14802 |
This theorem is referenced by: tanregt0 24285 atantan 24650 |
Copyright terms: Public domain | W3C validator |