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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 9020 | . . . 4 ⊢ 2 ∈ ℝ | |
2 | 2pos 9041 | . . . 4 ⊢ 0 < 2 | |
3 | 1 | leidi 8473 | . . . 4 ⊢ 2 ≤ 2 |
4 | 0xr 8035 | . . . . 5 ⊢ 0 ∈ ℝ* | |
5 | elioc2 9968 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 2 ∈ ℝ) → (2 ∈ (0(,]2) ↔ (2 ∈ ℝ ∧ 0 < 2 ∧ 2 ≤ 2))) | |
6 | 4, 1, 5 | mp2an 426 | . . . 4 ⊢ (2 ∈ (0(,]2) ↔ (2 ∈ ℝ ∧ 0 < 2 ∧ 2 ≤ 2)) |
7 | 1, 2, 3, 6 | mpbir3an 1181 | . . 3 ⊢ 2 ∈ (0(,]2) |
8 | sin02gt0 11806 | . . 3 ⊢ (2 ∈ (0(,]2) → 0 < (sin‘2)) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ 0 < (sin‘2) |
10 | cos2bnd 11803 | . . . 4 ⊢ (-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9)) | |
11 | 10 | simpri 113 | . . 3 ⊢ (cos‘2) < -(1 / 9) |
12 | 9re 9037 | . . . . 5 ⊢ 9 ∈ ℝ | |
13 | 9pos 9054 | . . . . 5 ⊢ 0 < 9 | |
14 | 12, 13 | recgt0ii 8895 | . . . 4 ⊢ 0 < (1 / 9) |
15 | 12, 13 | gt0ap0ii 8616 | . . . . . 6 ⊢ 9 # 0 |
16 | 12, 15 | rerecclapi 8765 | . . . . 5 ⊢ (1 / 9) ∈ ℝ |
17 | lt0neg2 8457 | . . . . 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 145 | . . 3 ⊢ -(1 / 9) < 0 |
20 | recoscl 11764 | . . . . 5 ⊢ (2 ∈ ℝ → (cos‘2) ∈ ℝ) | |
21 | 1, 20 | ax-mp 5 | . . . 4 ⊢ (cos‘2) ∈ ℝ |
22 | 16 | renegcli 8250 | . . . 4 ⊢ -(1 / 9) ∈ ℝ |
23 | 0re 7988 | . . . 4 ⊢ 0 ∈ ℝ | |
24 | 21, 22, 23 | lttri 8093 | . . 3 ⊢ (((cos‘2) < -(1 / 9) ∧ -(1 / 9) < 0) → (cos‘2) < 0) |
25 | 11, 19, 24 | mp2an 426 | . 2 ⊢ (cos‘2) < 0 |
26 | 9, 25 | pm3.2i 272 | 1 ⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∧ w3a 980 ∈ wcel 2160 class class class wbr 4018 ‘cfv 5235 (class class class)co 5897 ℝcr 7841 0cc0 7842 1c1 7843 ℝ*cxr 8022 < clt 8023 ≤ cle 8024 -cneg 8160 / cdiv 8660 2c2 9001 7c7 9006 9c9 9008 (,]cioc 9921 sincsin 11687 cosccos 11688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 ax-arch 7961 ax-caucvg 7962 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-disj 3996 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-irdg 6396 df-frec 6417 df-1o 6442 df-oadd 6446 df-er 6560 df-en 6768 df-dom 6769 df-fin 6770 df-sup 7014 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-5 9012 df-6 9013 df-7 9014 df-8 9015 df-9 9016 df-n0 9208 df-z 9285 df-uz 9560 df-q 9652 df-rp 9686 df-ioc 9925 df-ico 9926 df-fz 10041 df-fzo 10175 df-seqfrec 10479 df-exp 10554 df-fac 10741 df-bc 10763 df-ihash 10791 df-shft 10859 df-cj 10886 df-re 10887 df-im 10888 df-rsqrt 11042 df-abs 11043 df-clim 11322 df-sumdc 11397 df-ef 11691 df-sin 11693 df-cos 11694 |
This theorem is referenced by: sin4lt0 11809 cosz12 14678 |
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