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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 11789 | . . . 4 ⊢ (2 · 2) = 4 | |
2 | 1 | fveq2i 6666 | . . 3 ⊢ (sin‘(2 · 2)) = (sin‘4) |
3 | 2cn 11700 | . . . 4 ⊢ 2 ∈ ℂ | |
4 | sin2t 15518 | . . . 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 2843 | . 2 ⊢ (sin‘4) = (2 · ((sin‘2) · (cos‘2))) |
7 | sincos2sgn 15535 | . . . . . . 7 ⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) | |
8 | 7 | simpri 486 | . . . . . 6 ⊢ (cos‘2) < 0 |
9 | 2re 11699 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
10 | recoscl 15482 | . . . . . . . 8 ⊢ (2 ∈ ℝ → (cos‘2) ∈ ℝ) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ (cos‘2) ∈ ℝ |
12 | 0re 10631 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
13 | resincl 15481 | . . . . . . . . 9 ⊢ (2 ∈ ℝ → (sin‘2) ∈ ℝ) | |
14 | 9, 13 | ax-mp 5 | . . . . . . . 8 ⊢ (sin‘2) ∈ ℝ |
15 | 7 | simpli 484 | . . . . . . . 8 ⊢ 0 < (sin‘2) |
16 | 14, 15 | pm3.2i 471 | . . . . . . 7 ⊢ ((sin‘2) ∈ ℝ ∧ 0 < (sin‘2)) |
17 | ltmul2 11479 | . . . . . . 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 1452 | . . . . . 6 ⊢ ((cos‘2) < 0 ↔ ((sin‘2) · (cos‘2)) < ((sin‘2) · 0)) |
19 | 8, 18 | mpbi 231 | . . . . 5 ⊢ ((sin‘2) · (cos‘2)) < ((sin‘2) · 0) |
20 | 14 | recni 10643 | . . . . . 6 ⊢ (sin‘2) ∈ ℂ |
21 | 20 | mul01i 10818 | . . . . 5 ⊢ ((sin‘2) · 0) = 0 |
22 | 19, 21 | breqtri 5082 | . . . 4 ⊢ ((sin‘2) · (cos‘2)) < 0 |
23 | 14, 11 | remulcli 10645 | . . . . 5 ⊢ ((sin‘2) · (cos‘2)) ∈ ℝ |
24 | 2pos 11728 | . . . . . 6 ⊢ 0 < 2 | |
25 | 9, 24 | pm3.2i 471 | . . . . 5 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
26 | ltmul2 11479 | . . . . 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 1452 | . . . 4 ⊢ (((sin‘2) · (cos‘2)) < 0 ↔ (2 · ((sin‘2) · (cos‘2))) < (2 · 0)) |
28 | 22, 27 | mpbi 231 | . . 3 ⊢ (2 · ((sin‘2) · (cos‘2))) < (2 · 0) |
29 | 3 | mul01i 10818 | . . 3 ⊢ (2 · 0) = 0 |
30 | 28, 29 | breqtri 5082 | . 2 ⊢ (2 · ((sin‘2) · (cos‘2))) < 0 |
31 | 6, 30 | eqbrtri 5078 | 1 ⊢ (sin‘4) < 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 · cmul 10530 < clt 10663 2c2 11680 4c4 11682 sincsin 15405 cosccos 15406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ioc 12731 df-ico 12732 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 |
This theorem is referenced by: pilem3 24968 |
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