Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sincos2sgn | Structured version Visualization version 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 12058 | . . . 4 ⊢ 2 ∈ ℝ | |
2 | 2pos 12087 | . . . 4 ⊢ 0 < 2 | |
3 | 1 | leidi 11520 | . . . 4 ⊢ 2 ≤ 2 |
4 | 0xr 11033 | . . . . 5 ⊢ 0 ∈ ℝ* | |
5 | elioc2 13153 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 2 ∈ ℝ) → (2 ∈ (0(,]2) ↔ (2 ∈ ℝ ∧ 0 < 2 ∧ 2 ≤ 2))) | |
6 | 4, 1, 5 | mp2an 689 | . . . 4 ⊢ (2 ∈ (0(,]2) ↔ (2 ∈ ℝ ∧ 0 < 2 ∧ 2 ≤ 2)) |
7 | 1, 2, 3, 6 | mpbir3an 1340 | . . 3 ⊢ 2 ∈ (0(,]2) |
8 | sin02gt0 15912 | . . 3 ⊢ (2 ∈ (0(,]2) → 0 < (sin‘2)) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ 0 < (sin‘2) |
10 | cos2bnd 15908 | . . . 4 ⊢ (-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9)) | |
11 | 10 | simpri 486 | . . 3 ⊢ (cos‘2) < -(1 / 9) |
12 | 9re 12083 | . . . . 5 ⊢ 9 ∈ ℝ | |
13 | 9pos 12097 | . . . . 5 ⊢ 0 < 9 | |
14 | 12, 13 | recgt0ii 11892 | . . . 4 ⊢ 0 < (1 / 9) |
15 | 12, 13 | gt0ne0ii 11522 | . . . . . 6 ⊢ 9 ≠ 0 |
16 | 12, 15 | rereccli 11751 | . . . . 5 ⊢ (1 / 9) ∈ ℝ |
17 | lt0neg2 11493 | . . . . 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 229 | . . 3 ⊢ -(1 / 9) < 0 |
20 | recoscl 15861 | . . . . 5 ⊢ (2 ∈ ℝ → (cos‘2) ∈ ℝ) | |
21 | 1, 20 | ax-mp 5 | . . . 4 ⊢ (cos‘2) ∈ ℝ |
22 | 16 | renegcli 11293 | . . . 4 ⊢ -(1 / 9) ∈ ℝ |
23 | 0re 10988 | . . . 4 ⊢ 0 ∈ ℝ | |
24 | 21, 22, 23 | lttri 11112 | . . 3 ⊢ (((cos‘2) < -(1 / 9) ∧ -(1 / 9) < 0) → (cos‘2) < 0) |
25 | 11, 19, 24 | mp2an 689 | . 2 ⊢ (cos‘2) < 0 |
26 | 9, 25 | pm3.2i 471 | 1 ⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2110 class class class wbr 5079 ‘cfv 6432 (class class class)co 7272 ℝcr 10881 0cc0 10882 1c1 10883 ℝ*cxr 11019 < clt 11020 ≤ cle 11021 -cneg 11217 / cdiv 11643 2c2 12039 7c7 12044 9c9 12046 (,]cioc 13091 sincsin 15784 cosccos 15785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-inf2 9387 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-er 8490 df-pm 8610 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-sup 9189 df-inf 9190 df-oi 9257 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-uz 12594 df-rp 12742 df-ioc 13095 df-ico 13096 df-fz 13251 df-fzo 13394 df-fl 13523 df-seq 13733 df-exp 13794 df-fac 13999 df-bc 14028 df-hash 14056 df-shft 14789 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-limsup 15191 df-clim 15208 df-rlim 15209 df-sum 15409 df-ef 15788 df-sin 15790 df-cos 15791 |
This theorem is referenced by: sin4lt0 15915 pilem3 25623 |
Copyright terms: Public domain | W3C validator |