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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 11651 | . . . 4 ⊢ (2 · 2) = 4 | |
2 | 1 | fveq2i 6544 | . . 3 ⊢ (sin‘(2 · 2)) = (sin‘4) |
3 | 2cn 11562 | . . . 4 ⊢ 2 ∈ ℂ | |
4 | sin2t 15363 | . . . 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 2820 | . 2 ⊢ (sin‘4) = (2 · ((sin‘2) · (cos‘2))) |
7 | sincos2sgn 15380 | . . . . . . 7 ⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) | |
8 | 7 | simpri 486 | . . . . . 6 ⊢ (cos‘2) < 0 |
9 | 2re 11561 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
10 | recoscl 15327 | . . . . . . . 8 ⊢ (2 ∈ ℝ → (cos‘2) ∈ ℝ) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ (cos‘2) ∈ ℝ |
12 | 0re 10492 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
13 | resincl 15326 | . . . . . . . . 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 11341 | . . . . . . 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 1453 | . . . . . 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 10504 | . . . . . 6 ⊢ (sin‘2) ∈ ℂ |
21 | 20 | mul01i 10679 | . . . . 5 ⊢ ((sin‘2) · 0) = 0 |
22 | 19, 21 | breqtri 4989 | . . . 4 ⊢ ((sin‘2) · (cos‘2)) < 0 |
23 | 14, 11 | remulcli 10506 | . . . . 5 ⊢ ((sin‘2) · (cos‘2)) ∈ ℝ |
24 | 2pos 11590 | . . . . . 6 ⊢ 0 < 2 | |
25 | 9, 24 | pm3.2i 471 | . . . . 5 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
26 | ltmul2 11341 | . . . . 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 1453 | . . . 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 10679 | . . 3 ⊢ (2 · 0) = 0 |
30 | 28, 29 | breqtri 4989 | . 2 ⊢ (2 · ((sin‘2) · (cos‘2))) < 0 |
31 | 6, 30 | eqbrtri 4985 | 1 ⊢ (sin‘4) < 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2080 class class class wbr 4964 ‘cfv 6228 (class class class)co 7019 ℂcc 10384 ℝcr 10385 0cc0 10386 · cmul 10391 < clt 10524 2c2 11542 4c4 11544 sincsin 15250 cosccos 15251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-inf2 8953 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 ax-pre-sup 10464 ax-addf 10465 ax-mulf 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-se 5406 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-isom 6237 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-oadd 7960 df-er 8142 df-pm 8262 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-sup 8755 df-inf 8756 df-oi 8823 df-card 9217 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-div 11148 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-5 11553 df-6 11554 df-7 11555 df-8 11556 df-9 11557 df-n0 11748 df-z 11832 df-uz 12094 df-rp 12240 df-ioc 12593 df-ico 12594 df-fz 12743 df-fzo 12884 df-fl 13012 df-seq 13220 df-exp 13280 df-fac 13484 df-bc 13513 df-hash 13541 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 df-sin 15256 df-cos 15257 |
This theorem is referenced by: pilem3 24724 |
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