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Mirrors > Home > MPE Home > Th. List > sin4lt0 | Structured version Visualization version GIF version |
Description: The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.) |
Ref | Expression |
---|---|
sin4lt0 | ⊢ (sin‘4) < 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2t2e4 12324 | . . . 4 ⊢ (2 · 2) = 4 | |
2 | 1 | fveq2i 6850 | . . 3 ⊢ (sin‘(2 · 2)) = (sin‘4) |
3 | 2cn 12235 | . . . 4 ⊢ 2 ∈ ℂ | |
4 | sin2t 16066 | . . . 4 ⊢ (2 ∈ ℂ → (sin‘(2 · 2)) = (2 · ((sin‘2) · (cos‘2)))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (sin‘(2 · 2)) = (2 · ((sin‘2) · (cos‘2))) |
6 | 2, 5 | eqtr3i 2767 | . 2 ⊢ (sin‘4) = (2 · ((sin‘2) · (cos‘2))) |
7 | sincos2sgn 16083 | . . . . . . 7 ⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) | |
8 | 7 | simpri 487 | . . . . . 6 ⊢ (cos‘2) < 0 |
9 | 2re 12234 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
10 | recoscl 16030 | . . . . . . . 8 ⊢ (2 ∈ ℝ → (cos‘2) ∈ ℝ) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ (cos‘2) ∈ ℝ |
12 | 0re 11164 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
13 | resincl 16029 | . . . . . . . . 9 ⊢ (2 ∈ ℝ → (sin‘2) ∈ ℝ) | |
14 | 9, 13 | ax-mp 5 | . . . . . . . 8 ⊢ (sin‘2) ∈ ℝ |
15 | 7 | simpli 485 | . . . . . . . 8 ⊢ 0 < (sin‘2) |
16 | 14, 15 | pm3.2i 472 | . . . . . . 7 ⊢ ((sin‘2) ∈ ℝ ∧ 0 < (sin‘2)) |
17 | ltmul2 12013 | . . . . . . 7 ⊢ (((cos‘2) ∈ ℝ ∧ 0 ∈ ℝ ∧ ((sin‘2) ∈ ℝ ∧ 0 < (sin‘2))) → ((cos‘2) < 0 ↔ ((sin‘2) · (cos‘2)) < ((sin‘2) · 0))) | |
18 | 11, 12, 16, 17 | mp3an 1462 | . . . . . 6 ⊢ ((cos‘2) < 0 ↔ ((sin‘2) · (cos‘2)) < ((sin‘2) · 0)) |
19 | 8, 18 | mpbi 229 | . . . . 5 ⊢ ((sin‘2) · (cos‘2)) < ((sin‘2) · 0) |
20 | 14 | recni 11176 | . . . . . 6 ⊢ (sin‘2) ∈ ℂ |
21 | 20 | mul01i 11352 | . . . . 5 ⊢ ((sin‘2) · 0) = 0 |
22 | 19, 21 | breqtri 5135 | . . . 4 ⊢ ((sin‘2) · (cos‘2)) < 0 |
23 | 14, 11 | remulcli 11178 | . . . . 5 ⊢ ((sin‘2) · (cos‘2)) ∈ ℝ |
24 | 2pos 12263 | . . . . . 6 ⊢ 0 < 2 | |
25 | 9, 24 | pm3.2i 472 | . . . . 5 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
26 | ltmul2 12013 | . . . . 5 ⊢ ((((sin‘2) · (cos‘2)) ∈ ℝ ∧ 0 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((sin‘2) · (cos‘2)) < 0 ↔ (2 · ((sin‘2) · (cos‘2))) < (2 · 0))) | |
27 | 23, 12, 25, 26 | mp3an 1462 | . . . 4 ⊢ (((sin‘2) · (cos‘2)) < 0 ↔ (2 · ((sin‘2) · (cos‘2))) < (2 · 0)) |
28 | 22, 27 | mpbi 229 | . . 3 ⊢ (2 · ((sin‘2) · (cos‘2))) < (2 · 0) |
29 | 3 | mul01i 11352 | . . 3 ⊢ (2 · 0) = 0 |
30 | 28, 29 | breqtri 5135 | . 2 ⊢ (2 · ((sin‘2) · (cos‘2))) < 0 |
31 | 6, 30 | eqbrtri 5131 | 1 ⊢ (sin‘4) < 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 ℂcc 11056 ℝcr 11057 0cc0 11058 · cmul 11063 < clt 11196 2c2 12215 4c4 12217 sincsin 15953 cosccos 15954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-ioc 13276 df-ico 13277 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14959 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-sum 15578 df-ef 15957 df-sin 15959 df-cos 15960 |
This theorem is referenced by: pilem3 25828 |
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