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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 11877 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | 1 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → 𝐴 ∈ ℝ) |
3 | 2 | resincld 14917 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → (sin‘𝐴) ∈ ℝ) |
4 | 1red 10093 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → 1 ∈ ℝ) | |
5 | sinbnd 14954 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (-1 ≤ (sin‘𝐴) ∧ (sin‘𝐴) ≤ 1)) | |
6 | 5 | simprd 478 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) ≤ 1) |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (sin‘𝐴) ≤ 1) |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → (sin‘𝐴) ≤ 1) |
9 | simpr 476 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → 1 < 𝐴) | |
10 | 3, 4, 2, 8, 9 | lelttrd 10233 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 < 𝐴) → (sin‘𝐴) < 𝐴) |
11 | df-3an 1056 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1) ↔ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1)) | |
12 | 0xr 10124 | . . . . 5 ⊢ 0 ∈ ℝ* | |
13 | 1re 10077 | . . . . 5 ⊢ 1 ∈ ℝ | |
14 | elioc2 12274 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1))) | |
15 | 12, 13, 14 | mp2an 708 | . . . 4 ⊢ (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1)) |
16 | elrp 11872 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
17 | 16 | anbi1i 731 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ↔ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 𝐴 ≤ 1)) |
18 | 11, 15, 17 | 3bitr4i 292 | . . 3 ⊢ (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1)) |
19 | sin01bnd 14959 | . . . 4 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴 − ((𝐴↑3) / 3)) < (sin‘𝐴) ∧ (sin‘𝐴) < 𝐴)) | |
20 | 19 | simprd 478 | . . 3 ⊢ (𝐴 ∈ (0(,]1) → (sin‘𝐴) < 𝐴) |
21 | 18, 20 | sylbir 225 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (sin‘𝐴) < 𝐴) |
22 | 1red 10093 | . 2 ⊢ (𝐴 ∈ ℝ+ → 1 ∈ ℝ) | |
23 | 10, 21, 22, 1 | ltlecasei 10183 | 1 ⊢ (𝐴 ∈ ℝ+ → (sin‘𝐴) < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 ∈ wcel 2030 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 0cc0 9974 1c1 9975 ℝ*cxr 10111 < clt 10112 ≤ cle 10113 − cmin 10304 -cneg 10305 / cdiv 10722 3c3 11109 ℝ+crp 11870 (,]cioc 12214 ↑cexp 12900 sincsin 14838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-ioc 12218 df-ico 12219 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-sin 14844 df-cos 14845 |
This theorem is referenced by: basellem8 24859 pigt3 33532 |
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