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Mirrors > Home > ILE Home > Th. List > sincos2sgn | GIF version |
Description: The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
Ref | Expression |
---|---|
sincos2sgn | ⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8941 | . . . 4 ⊢ 2 ∈ ℝ | |
2 | 2pos 8962 | . . . 4 ⊢ 0 < 2 | |
3 | 1 | leidi 8397 | . . . 4 ⊢ 2 ≤ 2 |
4 | 0xr 7959 | . . . . 5 ⊢ 0 ∈ ℝ* | |
5 | elioc2 9886 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 2 ∈ ℝ) → (2 ∈ (0(,]2) ↔ (2 ∈ ℝ ∧ 0 < 2 ∧ 2 ≤ 2))) | |
6 | 4, 1, 5 | mp2an 424 | . . . 4 ⊢ (2 ∈ (0(,]2) ↔ (2 ∈ ℝ ∧ 0 < 2 ∧ 2 ≤ 2)) |
7 | 1, 2, 3, 6 | mpbir3an 1174 | . . 3 ⊢ 2 ∈ (0(,]2) |
8 | sin02gt0 11719 | . . 3 ⊢ (2 ∈ (0(,]2) → 0 < (sin‘2)) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ 0 < (sin‘2) |
10 | cos2bnd 11716 | . . . 4 ⊢ (-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9)) | |
11 | 10 | simpri 112 | . . 3 ⊢ (cos‘2) < -(1 / 9) |
12 | 9re 8958 | . . . . 5 ⊢ 9 ∈ ℝ | |
13 | 9pos 8975 | . . . . 5 ⊢ 0 < 9 | |
14 | 12, 13 | recgt0ii 8816 | . . . 4 ⊢ 0 < (1 / 9) |
15 | 12, 13 | gt0ap0ii 8540 | . . . . . 6 ⊢ 9 # 0 |
16 | 12, 15 | rerecclapi 8687 | . . . . 5 ⊢ (1 / 9) ∈ ℝ |
17 | lt0neg2 8381 | . . . . 5 ⊢ ((1 / 9) ∈ ℝ → (0 < (1 / 9) ↔ -(1 / 9) < 0)) | |
18 | 16, 17 | ax-mp 5 | . . . 4 ⊢ (0 < (1 / 9) ↔ -(1 / 9) < 0) |
19 | 14, 18 | mpbi 144 | . . 3 ⊢ -(1 / 9) < 0 |
20 | recoscl 11677 | . . . . 5 ⊢ (2 ∈ ℝ → (cos‘2) ∈ ℝ) | |
21 | 1, 20 | ax-mp 5 | . . . 4 ⊢ (cos‘2) ∈ ℝ |
22 | 16 | renegcli 8174 | . . . 4 ⊢ -(1 / 9) ∈ ℝ |
23 | 0re 7913 | . . . 4 ⊢ 0 ∈ ℝ | |
24 | 21, 22, 23 | lttri 8017 | . . 3 ⊢ (((cos‘2) < -(1 / 9) ∧ -(1 / 9) < 0) → (cos‘2) < 0) |
25 | 11, 19, 24 | mp2an 424 | . 2 ⊢ (cos‘2) < 0 |
26 | 9, 25 | pm3.2i 270 | 1 ⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∧ w3a 973 ∈ wcel 2141 class class class wbr 3987 ‘cfv 5196 (class class class)co 5851 ℝcr 7766 0cc0 7767 1c1 7768 ℝ*cxr 7946 < clt 7947 ≤ cle 7948 -cneg 8084 / cdiv 8582 2c2 8922 7c7 8927 9c9 8929 (,]cioc 9839 sincsin 11600 cosccos 11601 |
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 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 ax-arch 7886 ax-caucvg 7887 |
This theorem depends on definitions: df-bi 116 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 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-disj 3965 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-frec 6368 df-1o 6393 df-oadd 6397 df-er 6511 df-en 6717 df-dom 6718 df-fin 6719 df-sup 6959 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-3 8931 df-4 8932 df-5 8933 df-6 8934 df-7 8935 df-8 8936 df-9 8937 df-n0 9129 df-z 9206 df-uz 9481 df-q 9572 df-rp 9604 df-ioc 9843 df-ico 9844 df-fz 9959 df-fzo 10092 df-seqfrec 10395 df-exp 10469 df-fac 10653 df-bc 10675 df-ihash 10703 df-shft 10772 df-cj 10799 df-re 10800 df-im 10801 df-rsqrt 10955 df-abs 10956 df-clim 11235 df-sumdc 11310 df-ef 11604 df-sin 11606 df-cos 11607 |
This theorem is referenced by: sin4lt0 11722 cosz12 13460 |
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