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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 14728 | . 2 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ ℝ) | |
2 | ax-icn 9874 | . . . . . . . 8 ⊢ i ∈ ℂ | |
3 | recn 9905 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | mulcl 9899 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 694 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
6 | rpcoshcl 14726 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ∈ ℝ+) | |
7 | 6 | rpne0d 11753 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (cos‘(i · 𝐴)) ≠ 0) |
8 | 5, 7 | tancld 14701 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (tan‘(i · 𝐴)) ∈ ℂ) |
9 | 2 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → i ∈ ℂ) |
10 | ine0 10344 | . . . . . . 7 ⊢ i ≠ 0 | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → i ≠ 0) |
12 | 8, 9, 11 | divnegd 10693 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) = (-(tan‘(i · 𝐴)) / i)) |
13 | mulneg2 10346 | . . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴)) | |
14 | 2, 3, 13 | sylancr 694 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (i · -𝐴) = -(i · 𝐴)) |
15 | 14 | fveq2d 6107 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (tan‘(i · -𝐴)) = (tan‘-(i · 𝐴))) |
16 | tanneg 14717 | . . . . . . . 8 ⊢ (((i · 𝐴) ∈ ℂ ∧ (cos‘(i · 𝐴)) ≠ 0) → (tan‘-(i · 𝐴)) = -(tan‘(i · 𝐴))) | |
17 | 5, 7, 16 | syl2anc 691 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (tan‘-(i · 𝐴)) = -(tan‘(i · 𝐴))) |
18 | 15, 17 | eqtrd 2644 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (tan‘(i · -𝐴)) = -(tan‘(i · 𝐴))) |
19 | 18 | oveq1d 6564 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · -𝐴)) / i) = (-(tan‘(i · 𝐴)) / i)) |
20 | 12, 19 | eqtr4d 2647 | . . . 4 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) = ((tan‘(i · -𝐴)) / i)) |
21 | renegcl 10223 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
22 | tanhlt1 14729 | . . . . 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 4605 | . . 3 ⊢ (𝐴 ∈ ℝ → -((tan‘(i · 𝐴)) / i) < 1) |
25 | 1re 9918 | . . . 4 ⊢ 1 ∈ ℝ | |
26 | ltnegcon1 10408 | . . . 4 ⊢ ((((tan‘(i · 𝐴)) / i) ∈ ℝ ∧ 1 ∈ ℝ) → (-((tan‘(i · 𝐴)) / i) < 1 ↔ -1 < ((tan‘(i · 𝐴)) / i))) | |
27 | 1, 25, 26 | sylancl 693 | . . 3 ⊢ (𝐴 ∈ ℝ → (-((tan‘(i · 𝐴)) / i) < 1 ↔ -1 < ((tan‘(i · 𝐴)) / i))) |
28 | 24, 27 | mpbid 221 | . 2 ⊢ (𝐴 ∈ ℝ → -1 < ((tan‘(i · 𝐴)) / i)) |
29 | tanhlt1 14729 | . 2 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) < 1) | |
30 | neg1rr 11002 | . . . 4 ⊢ -1 ∈ ℝ | |
31 | 30 | rexri 9976 | . . 3 ⊢ -1 ∈ ℝ* |
32 | 25 | rexri 9976 | . . 3 ⊢ 1 ∈ ℝ* |
33 | elioo2 12087 | . . 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 704 | . 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 1239 | 1 ⊢ (𝐴 ∈ ℝ → ((tan‘(i · 𝐴)) / i) ∈ (-1(,)1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 ici 9817 · cmul 9820 ℝ*cxr 9952 < clt 9953 -cneg 10146 / cdiv 10563 (,)cioo 12046 cosccos 14634 tanctan 14635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ioo 12050 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-tan 14641 |
This theorem is referenced by: tanregt0 24089 atantan 24450 |
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