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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 | β’ (π΄ β β β ((tanβ(i Β· π΄)) / i) β (-1(,)1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retanhcl 16143 | . 2 β’ (π΄ β β β ((tanβ(i Β· π΄)) / i) β β) | |
2 | ax-icn 11205 | . . . . . . . 8 β’ i β β | |
3 | recn 11236 | . . . . . . . 8 β’ (π΄ β β β π΄ β β) | |
4 | mulcl 11230 | . . . . . . . 8 β’ ((i β β β§ π΄ β β) β (i Β· π΄) β β) | |
5 | 2, 3, 4 | sylancr 585 | . . . . . . 7 β’ (π΄ β β β (i Β· π΄) β β) |
6 | rpcoshcl 16141 | . . . . . . . 8 β’ (π΄ β β β (cosβ(i Β· π΄)) β β+) | |
7 | 6 | rpne0d 13061 | . . . . . . 7 β’ (π΄ β β β (cosβ(i Β· π΄)) β 0) |
8 | 5, 7 | tancld 16116 | . . . . . 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 11687 | . . . . . . 7 β’ i β 0 | |
11 | 10 | a1i 11 | . . . . . 6 β’ (π΄ β β β i β 0) |
12 | 8, 9, 11 | divnegd 12041 | . . . . 5 β’ (π΄ β β β -((tanβ(i Β· π΄)) / i) = (-(tanβ(i Β· π΄)) / i)) |
13 | mulneg2 11689 | . . . . . . . . 9 β’ ((i β β β§ π΄ β β) β (i Β· -π΄) = -(i Β· π΄)) | |
14 | 2, 3, 13 | sylancr 585 | . . . . . . . 8 β’ (π΄ β β β (i Β· -π΄) = -(i Β· π΄)) |
15 | 14 | fveq2d 6906 | . . . . . . 7 β’ (π΄ β β β (tanβ(i Β· -π΄)) = (tanβ-(i Β· π΄))) |
16 | tanneg 16132 | . . . . . . . 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 582 | . . . . . . 7 β’ (π΄ β β β (tanβ-(i Β· π΄)) = -(tanβ(i Β· π΄))) |
18 | 15, 17 | eqtrd 2768 | . . . . . 6 β’ (π΄ β β β (tanβ(i Β· -π΄)) = -(tanβ(i Β· π΄))) |
19 | 18 | oveq1d 7441 | . . . . 5 β’ (π΄ β β β ((tanβ(i Β· -π΄)) / i) = (-(tanβ(i Β· π΄)) / i)) |
20 | 12, 19 | eqtr4d 2771 | . . . 4 β’ (π΄ β β β -((tanβ(i Β· π΄)) / i) = ((tanβ(i Β· -π΄)) / i)) |
21 | renegcl 11561 | . . . . 5 β’ (π΄ β β β -π΄ β β) | |
22 | tanhlt1 16144 | . . . . 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 5174 | . . 3 β’ (π΄ β β β -((tanβ(i Β· π΄)) / i) < 1) |
25 | 1re 11252 | . . . 4 β’ 1 β β | |
26 | ltnegcon1 11753 | . . . 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 584 | . . 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 16144 | . 2 β’ (π΄ β β β ((tanβ(i Β· π΄)) / i) < 1) | |
30 | neg1rr 12365 | . . . 4 β’ -1 β β | |
31 | 30 | rexri 11310 | . . 3 β’ -1 β β* |
32 | 25 | rexri 11310 | . . 3 β’ 1 β β* |
33 | elioo2 13405 | . . 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 690 | . 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 2937 class class class wbr 5152 βcfv 6553 (class class class)co 7426 βcc 11144 βcr 11145 0cc0 11146 1c1 11147 ici 11148 Β· cmul 11151 β*cxr 11285 < clt 11286 -cneg 11483 / cdiv 11909 (,)cioo 13364 cosccos 16048 tanctan 16049 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-ioo 13368 df-ico 13370 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 df-tan 16055 |
This theorem is referenced by: tanregt0 26493 atantan 26875 |
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