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Mirrors > Home > MPE Home > Th. List > sinltx | Structured version Visualization version GIF version |
Description: The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.) |
Ref | Expression |
---|---|
sinltx | ⊢ (𝐴 ∈ ℝ+ → (sin‘𝐴) < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 12120 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | 1 | adantr 474 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → 𝐴 ∈ ℝ) |
3 | 2 | resincld 15245 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → (sin‘𝐴) ∈ ℝ) |
4 | 1red 10357 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → 1 ∈ ℝ) | |
5 | sinbnd 15282 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (-1 ≤ (sin‘𝐴) ∧ (sin‘𝐴) ≤ 1)) | |
6 | 5 | simprd 491 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ≤ 1) |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (sin‘𝐴) ≤ 1) |
8 | 7 | adantr 474 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → (sin‘𝐴) ≤ 1) |
9 | simpr 479 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → 1 < 𝐴) | |
10 | 3, 4, 2, 8, 9 | lelttrd 10514 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → (sin‘𝐴) < 𝐴) |
11 | df-3an 1115 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1) ↔ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1)) | |
12 | 0xr 10403 | . . . . 5 ⊢ 0 ∈ ℝ* | |
13 | 1re 10356 | . . . . 5 ⊢ 1 ∈ ℝ | |
14 | elioc2 12524 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1))) | |
15 | 12, 13, 14 | mp2an 685 | . . . 4 ⊢ (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1)) |
16 | elrp 12114 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
17 | 16 | anbi1i 619 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ↔ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1)) |
18 | 11, 15, 17 | 3bitr4i 295 | . . 3 ⊢ (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1)) |
19 | sin01bnd 15287 | . . . 4 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴 − ((𝐴↑3) / 3)) < (sin‘𝐴) ∧ (sin‘𝐴) < 𝐴)) | |
20 | 19 | simprd 491 | . . 3 ⊢ (𝐴 ∈ (0(,]1) → (sin‘𝐴) < 𝐴) |
21 | 18, 20 | sylbir 227 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (sin‘𝐴) < 𝐴) |
22 | 1red 10357 | . 2 ⊢ (𝐴 ∈ ℝ+ → 1 ∈ ℝ) | |
23 | 10, 21, 22, 1 | ltlecasei 10464 | 1 ⊢ (𝐴 ∈ ℝ+ → (sin‘𝐴) < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 ∈ wcel 2166 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 ℝcr 10251 0cc0 10252 1c1 10253 ℝ*cxr 10390 < clt 10391 ≤ cle 10392 − cmin 10585 -cneg 10586 / cdiv 11009 3c3 11407 ℝ+crp 12112 (,]cioc 12464 ↑cexp 13154 sincsin 15166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 ax-addf 10331 ax-mulf 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-ioc 12468 df-ico 12469 df-fz 12620 df-fzo 12761 df-fl 12888 df-seq 13096 df-exp 13155 df-fac 13354 df-bc 13383 df-hash 13411 df-shft 14184 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-limsup 14579 df-clim 14596 df-rlim 14597 df-sum 14794 df-ef 15170 df-sin 15172 df-cos 15173 |
This theorem is referenced by: basellem8 25227 pigt3 33945 |
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