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Mirrors > Home > ILE Home > Th. List > sincosq1lem | GIF version |
Description: Lemma for sincosq1sgn 13541. (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 13507 | . . . . . 6 ⊢ (π / 2) ∈ ℝ | |
2 | ltle 8007 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (𝐴 < (π / 2) → 𝐴 ≤ (π / 2))) | |
3 | 1, 2 | mpan2 423 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 < (π / 2) → 𝐴 ≤ (π / 2))) |
4 | pire 13501 | . . . . . . . 8 ⊢ π ∈ ℝ | |
5 | 4re 8955 | . . . . . . . 8 ⊢ 4 ∈ ℝ | |
6 | pigt2lt4 13499 | . . . . . . . . 9 ⊢ (2 < π ∧ π < 4) | |
7 | 6 | simpri 112 | . . . . . . . 8 ⊢ π < 4 |
8 | 4, 5, 7 | ltleii 8022 | . . . . . . 7 ⊢ π ≤ 4 |
9 | 2re 8948 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
10 | 2pos 8969 | . . . . . . . . . 10 ⊢ 0 < 2 | |
11 | 9, 10 | pm3.2i 270 | . . . . . . . . 9 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
12 | ledivmul 8793 | . . . . . . . . 9 ⊢ ((π ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((π / 2) ≤ 2 ↔ π ≤ (2 · 2))) | |
13 | 4, 9, 11, 12 | mp3an 1332 | . . . . . . . 8 ⊢ ((π / 2) ≤ 2 ↔ π ≤ (2 · 2)) |
14 | 2t2e4 9032 | . . . . . . . . 9 ⊢ (2 · 2) = 4 | |
15 | 14 | breq2i 3997 | . . . . . . . 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 8002 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ 2 ∈ ℝ) → ((𝐴 ≤ (π / 2) ∧ (π / 2) ≤ 2) → 𝐴 ≤ 2)) | |
19 | 1, 9, 18 | mp3an23 1324 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝐴 ≤ (π / 2) ∧ (π / 2) ≤ 2) → 𝐴 ≤ 2)) |
20 | 17, 19 | mpan2i 429 | . . . . 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 1195 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 𝐴 ≤ 2) |
24 | 0xr 7966 | . . . 4 ⊢ 0 ∈ ℝ* | |
25 | elioc2 9893 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ 2 ∈ ℝ) → (𝐴 ∈ (0(,]2) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 2))) | |
26 | 24, 9, 25 | mp2an 424 | . . 3 ⊢ (𝐴 ∈ (0(,]2) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 2)) |
27 | sin02gt0 11726 | . . 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 1278 | 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 973 ∈ wcel 2141 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 ℝcr 7773 0cc0 7774 · cmul 7779 ℝ*cxr 7953 < clt 7954 ≤ cle 7955 / cdiv 8589 2c2 8929 4c4 8931 (,]cioc 9846 sincsin 11607 πcpi 11610 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 ax-pre-suploc 7895 ax-addf 7896 ax-mulf 7897 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-disj 3967 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-of 6061 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-map 6628 df-pm 6629 df-en 6719 df-dom 6720 df-fin 6721 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-9 8944 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-xneg 9729 df-xadd 9730 df-ioo 9849 df-ioc 9850 df-ico 9851 df-icc 9852 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-fac 10660 df-bc 10682 df-ihash 10710 df-shft 10779 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317 df-ef 11611 df-sin 11613 df-cos 11614 df-pi 11616 df-rest 12581 df-topgen 12600 df-psmet 12781 df-xmet 12782 df-met 12783 df-bl 12784 df-mopn 12785 df-top 12790 df-topon 12803 df-bases 12835 df-ntr 12890 df-cn 12982 df-cnp 12983 df-tx 13047 df-cncf 13352 df-limced 13419 df-dvap 13420 |
This theorem is referenced by: sincosq1sgn 13541 sinq12gt0 13545 |
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