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Mirrors > Home > ILE Home > Th. List > sincosq1lem | GIF version |
Description: Lemma for sincosq1sgn 13294. (Contributed by Paul Chapman, 24-Jan-2008.) |
Ref | Expression |
---|---|
sincosq1lem | ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (sin‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfpire 13260 | . . . . . 6 ⊢ (π / 2) ∈ ℝ | |
2 | ltle 7977 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (𝐴 < (π / 2) → 𝐴 ≤ (π / 2))) | |
3 | 1, 2 | mpan2 422 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 < (π / 2) → 𝐴 ≤ (π / 2))) |
4 | pire 13254 | . . . . . . . 8 ⊢ π ∈ ℝ | |
5 | 4re 8925 | . . . . . . . 8 ⊢ 4 ∈ ℝ | |
6 | pigt2lt4 13252 | . . . . . . . . 9 ⊢ (2 < π ∧ π < 4) | |
7 | 6 | simpri 112 | . . . . . . . 8 ⊢ π < 4 |
8 | 4, 5, 7 | ltleii 7992 | . . . . . . 7 ⊢ π ≤ 4 |
9 | 2re 8918 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
10 | 2pos 8939 | . . . . . . . . . 10 ⊢ 0 < 2 | |
11 | 9, 10 | pm3.2i 270 | . . . . . . . . 9 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
12 | ledivmul 8763 | . . . . . . . . 9 ⊢ ((π ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((π / 2) ≤ 2 ↔ π ≤ (2 · 2))) | |
13 | 4, 9, 11, 12 | mp3an 1326 | . . . . . . . 8 ⊢ ((π / 2) ≤ 2 ↔ π ≤ (2 · 2)) |
14 | 2t2e4 9002 | . . . . . . . . 9 ⊢ (2 · 2) = 4 | |
15 | 14 | breq2i 3984 | . . . . . . . 8 ⊢ (π ≤ (2 · 2) ↔ π ≤ 4) |
16 | 13, 15 | bitri 183 | . . . . . . 7 ⊢ ((π / 2) ≤ 2 ↔ π ≤ 4) |
17 | 8, 16 | mpbir 145 | . . . . . 6 ⊢ (π / 2) ≤ 2 |
18 | letr 7972 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ 2 ∈ ℝ) → ((𝐴 ≤ (π / 2) ∧ (π / 2) ≤ 2) → 𝐴 ≤ 2)) | |
19 | 1, 9, 18 | mp3an23 1318 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝐴 ≤ (π / 2) ∧ (π / 2) ≤ 2) → 𝐴 ≤ 2)) |
20 | 17, 19 | mpan2i 428 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ (π / 2) → 𝐴 ≤ 2)) |
21 | 3, 20 | syld 45 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 < (π / 2) → 𝐴 ≤ 2)) |
22 | 21 | adantr 274 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 < (π / 2) → 𝐴 ≤ 2)) |
23 | 22 | 3impia 1189 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 𝐴 ≤ 2) |
24 | 0xr 7936 | . . . 4 ⊢ 0 ∈ ℝ* | |
25 | elioc2 9863 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ 2 ∈ ℝ) → (𝐴 ∈ (0(,]2) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 2))) | |
26 | 24, 9, 25 | mp2an 423 | . . 3 ⊢ (𝐴 ∈ (0(,]2) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 2)) |
27 | sin02gt0 11690 | . . 3 ⊢ (𝐴 ∈ (0(,]2) → 0 < (sin‘𝐴)) | |
28 | 26, 27 | sylbir 134 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 2) → 0 < (sin‘𝐴)) |
29 | 23, 28 | syld3an3 1272 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (sin‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 967 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℝcr 7743 0cc0 7744 · cmul 7749 ℝ*cxr 7923 < clt 7924 ≤ cle 7925 / cdiv 8559 2c2 8899 4c4 8901 (,]cioc 9816 sincsin 11571 πcpi 11574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 ax-pre-suploc 7865 ax-addf 7866 ax-mulf 7867 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-disj 3954 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-of 6044 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-map 6607 df-pm 6608 df-en 6698 df-dom 6699 df-fin 6700 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-xneg 9699 df-xadd 9700 df-ioo 9819 df-ioc 9820 df-ico 9821 df-icc 9822 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-fac 10628 df-bc 10650 df-ihash 10678 df-shft 10743 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281 df-ef 11575 df-sin 11577 df-cos 11578 df-pi 11580 df-rest 12500 df-topgen 12519 df-psmet 12534 df-xmet 12535 df-met 12536 df-bl 12537 df-mopn 12538 df-top 12543 df-topon 12556 df-bases 12588 df-ntr 12643 df-cn 12735 df-cnp 12736 df-tx 12800 df-cncf 13105 df-limced 13172 df-dvap 13173 |
This theorem is referenced by: sincosq1sgn 13294 sinq12gt0 13298 |
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