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Mirrors > Home > ILE Home > Th. List > sin4lt0 | 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 9002 | . . . 4 ⊢ (2 · 2) = 4 | |
2 | 1 | fveq2i 5483 | . . 3 ⊢ (sin‘(2 · 2)) = (sin‘4) |
3 | 2cn 8919 | . . . 4 ⊢ 2 ∈ ℂ | |
4 | sin2t 11676 | . . . 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 2187 | . 2 ⊢ (sin‘4) = (2 · ((sin‘2) · (cos‘2))) |
7 | sincos2sgn 11692 | . . . . . . 7 ⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) | |
8 | 7 | simpri 112 | . . . . . 6 ⊢ (cos‘2) < 0 |
9 | 2re 8918 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
10 | recoscl 11648 | . . . . . . . 8 ⊢ (2 ∈ ℝ → (cos‘2) ∈ ℝ) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ (cos‘2) ∈ ℝ |
12 | 0re 7890 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
13 | resincl 11647 | . . . . . . . . 9 ⊢ (2 ∈ ℝ → (sin‘2) ∈ ℝ) | |
14 | 9, 13 | ax-mp 5 | . . . . . . . 8 ⊢ (sin‘2) ∈ ℝ |
15 | 7 | simpli 110 | . . . . . . . 8 ⊢ 0 < (sin‘2) |
16 | 14, 15 | pm3.2i 270 | . . . . . . 7 ⊢ ((sin‘2) ∈ ℝ ∧ 0 < (sin‘2)) |
17 | ltmul2 8742 | . . . . . . 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 1326 | . . . . . 6 ⊢ ((cos‘2) < 0 ↔ ((sin‘2) · (cos‘2)) < ((sin‘2) · 0)) |
19 | 8, 18 | mpbi 144 | . . . . 5 ⊢ ((sin‘2) · (cos‘2)) < ((sin‘2) · 0) |
20 | 14 | recni 7902 | . . . . . 6 ⊢ (sin‘2) ∈ ℂ |
21 | 20 | mul01i 8280 | . . . . 5 ⊢ ((sin‘2) · 0) = 0 |
22 | 19, 21 | breqtri 4001 | . . . 4 ⊢ ((sin‘2) · (cos‘2)) < 0 |
23 | 14, 11 | remulcli 7904 | . . . . 5 ⊢ ((sin‘2) · (cos‘2)) ∈ ℝ |
24 | 2pos 8939 | . . . . . 6 ⊢ 0 < 2 | |
25 | 9, 24 | pm3.2i 270 | . . . . 5 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
26 | ltmul2 8742 | . . . . 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 1326 | . . . 4 ⊢ (((sin‘2) · (cos‘2)) < 0 ↔ (2 · ((sin‘2) · (cos‘2))) < (2 · 0)) |
28 | 22, 27 | mpbi 144 | . . 3 ⊢ (2 · ((sin‘2) · (cos‘2))) < (2 · 0) |
29 | 3 | mul01i 8280 | . . 3 ⊢ (2 · 0) = 0 |
30 | 28, 29 | breqtri 4001 | . 2 ⊢ (2 · ((sin‘2) · (cos‘2))) < 0 |
31 | 6, 30 | eqbrtri 3997 | 1 ⊢ (sin‘4) < 0 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℂcc 7742 ℝcr 7743 0cc0 7744 · cmul 7749 < clt 7924 2c2 8899 4c4 8901 sincsin 11571 cosccos 11572 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-disj 3954 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 df-sup 6940 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-ioc 9820 df-ico 9821 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-fac 10628 df-bc 10650 df-ihash 10678 df-shft 10743 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281 df-ef 11575 df-sin 11577 df-cos 11578 |
This theorem is referenced by: (None) |
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