![]() |
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 | β’ (π΄ β β β ((tanβ(i Β· π΄)) / i) β (-1(,)1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retanhcl 16109 | . 2 β’ (π΄ β β β ((tanβ(i Β· π΄)) / i) β β) | |
2 | ax-icn 11171 | . . . . . . . 8 β’ i β β | |
3 | recn 11202 | . . . . . . . 8 β’ (π΄ β β β π΄ β β) | |
4 | mulcl 11196 | . . . . . . . 8 β’ ((i β β β§ π΄ β β) β (i Β· π΄) β β) | |
5 | 2, 3, 4 | sylancr 586 | . . . . . . 7 β’ (π΄ β β β (i Β· π΄) β β) |
6 | rpcoshcl 16107 | . . . . . . . 8 β’ (π΄ β β β (cosβ(i Β· π΄)) β β+) | |
7 | 6 | rpne0d 13027 | . . . . . . 7 β’ (π΄ β β β (cosβ(i Β· π΄)) β 0) |
8 | 5, 7 | tancld 16082 | . . . . . 6 β’ (π΄ β β β (tanβ(i Β· π΄)) β β) |
9 | 2 | a1i 11 | . . . . . 6 β’ (π΄ β β β i β β) |
10 | ine0 11653 | . . . . . . 7 β’ i β 0 | |
11 | 10 | a1i 11 | . . . . . 6 β’ (π΄ β β β i β 0) |
12 | 8, 9, 11 | divnegd 12007 | . . . . 5 β’ (π΄ β β β -((tanβ(i Β· π΄)) / i) = (-(tanβ(i Β· π΄)) / i)) |
13 | mulneg2 11655 | . . . . . . . . 9 β’ ((i β β β§ π΄ β β) β (i Β· -π΄) = -(i Β· π΄)) | |
14 | 2, 3, 13 | sylancr 586 | . . . . . . . 8 β’ (π΄ β β β (i Β· -π΄) = -(i Β· π΄)) |
15 | 14 | fveq2d 6889 | . . . . . . 7 β’ (π΄ β β β (tanβ(i Β· -π΄)) = (tanβ-(i Β· π΄))) |
16 | tanneg 16098 | . . . . . . . 8 β’ (((i Β· π΄) β β β§ (cosβ(i Β· π΄)) β 0) β (tanβ-(i Β· π΄)) = -(tanβ(i Β· π΄))) | |
17 | 5, 7, 16 | syl2anc 583 | . . . . . . 7 β’ (π΄ β β β (tanβ-(i Β· π΄)) = -(tanβ(i Β· π΄))) |
18 | 15, 17 | eqtrd 2766 | . . . . . 6 β’ (π΄ β β β (tanβ(i Β· -π΄)) = -(tanβ(i Β· π΄))) |
19 | 18 | oveq1d 7420 | . . . . 5 β’ (π΄ β β β ((tanβ(i Β· -π΄)) / i) = (-(tanβ(i Β· π΄)) / i)) |
20 | 12, 19 | eqtr4d 2769 | . . . 4 β’ (π΄ β β β -((tanβ(i Β· π΄)) / i) = ((tanβ(i Β· -π΄)) / i)) |
21 | renegcl 11527 | . . . . 5 β’ (π΄ β β β -π΄ β β) | |
22 | tanhlt1 16110 | . . . . 5 β’ (-π΄ β β β ((tanβ(i Β· -π΄)) / i) < 1) | |
23 | 21, 22 | syl 17 | . . . 4 β’ (π΄ β β β ((tanβ(i Β· -π΄)) / i) < 1) |
24 | 20, 23 | eqbrtrd 5163 | . . 3 β’ (π΄ β β β -((tanβ(i Β· π΄)) / i) < 1) |
25 | 1re 11218 | . . . 4 β’ 1 β β | |
26 | ltnegcon1 11719 | . . . 4 β’ ((((tanβ(i Β· π΄)) / i) β β β§ 1 β β) β (-((tanβ(i Β· π΄)) / i) < 1 β -1 < ((tanβ(i Β· π΄)) / i))) | |
27 | 1, 25, 26 | sylancl 585 | . . 3 β’ (π΄ β β β (-((tanβ(i Β· π΄)) / i) < 1 β -1 < ((tanβ(i Β· π΄)) / i))) |
28 | 24, 27 | mpbid 231 | . 2 β’ (π΄ β β β -1 < ((tanβ(i Β· π΄)) / i)) |
29 | tanhlt1 16110 | . 2 β’ (π΄ β β β ((tanβ(i Β· π΄)) / i) < 1) | |
30 | neg1rr 12331 | . . . 4 β’ -1 β β | |
31 | 30 | rexri 11276 | . . 3 β’ -1 β β* |
32 | 25 | rexri 11276 | . . 3 β’ 1 β β* |
33 | elioo2 13371 | . . 3 β’ ((-1 β β* β§ 1 β β*) β (((tanβ(i Β· π΄)) / i) β (-1(,)1) β (((tanβ(i Β· π΄)) / i) β β β§ -1 < ((tanβ(i Β· π΄)) / i) β§ ((tanβ(i Β· π΄)) / i) < 1))) | |
34 | 31, 32, 33 | mp2an 689 | . 2 β’ (((tanβ(i Β· π΄)) / i) β (-1(,)1) β (((tanβ(i Β· π΄)) / i) β β β§ -1 < ((tanβ(i Β· π΄)) / i) β§ ((tanβ(i Β· π΄)) / i) < 1)) |
35 | 1, 28, 29, 34 | syl3anbrc 1340 | 1 β’ (π΄ β β β ((tanβ(i Β· π΄)) / i) β (-1(,)1)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 class class class wbr 5141 βcfv 6537 (class class class)co 7405 βcc 11110 βcr 11111 0cc0 11112 1c1 11113 ici 11114 Β· cmul 11117 β*cxr 11251 < clt 11252 -cneg 11449 / cdiv 11875 (,)cioo 13330 cosccos 16014 tanctan 16015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-ioo 13334 df-ico 13336 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-ef 16017 df-sin 16019 df-cos 16020 df-tan 16021 |
This theorem is referenced by: tanregt0 26428 atantan 26810 |
Copyright terms: Public domain | W3C validator |