![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 9102 | . . . 4 ⊢ (2 · 2) = 4 | |
2 | 1 | fveq2i 5537 | . . 3 ⊢ (sin‘(2 · 2)) = (sin‘4) |
3 | 2cn 9019 | . . . 4 ⊢ 2 ∈ ℂ | |
4 | sin2t 11788 | . . . 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 2212 | . 2 ⊢ (sin‘4) = (2 · ((sin‘2) · (cos‘2))) |
7 | sincos2sgn 11804 | . . . . . . 7 ⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) | |
8 | 7 | simpri 113 | . . . . . 6 ⊢ (cos‘2) < 0 |
9 | 2re 9018 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
10 | recoscl 11760 | . . . . . . . 8 ⊢ (2 ∈ ℝ → (cos‘2) ∈ ℝ) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ (cos‘2) ∈ ℝ |
12 | 0re 7986 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
13 | resincl 11759 | . . . . . . . . 9 ⊢ (2 ∈ ℝ → (sin‘2) ∈ ℝ) | |
14 | 9, 13 | ax-mp 5 | . . . . . . . 8 ⊢ (sin‘2) ∈ ℝ |
15 | 7 | simpli 111 | . . . . . . . 8 ⊢ 0 < (sin‘2) |
16 | 14, 15 | pm3.2i 272 | . . . . . . 7 ⊢ ((sin‘2) ∈ ℝ ∧ 0 < (sin‘2)) |
17 | ltmul2 8842 | . . . . . . 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 1348 | . . . . . 6 ⊢ ((cos‘2) < 0 ↔ ((sin‘2) · (cos‘2)) < ((sin‘2) · 0)) |
19 | 8, 18 | mpbi 145 | . . . . 5 ⊢ ((sin‘2) · (cos‘2)) < ((sin‘2) · 0) |
20 | 14 | recni 7998 | . . . . . 6 ⊢ (sin‘2) ∈ ℂ |
21 | 20 | mul01i 8377 | . . . . 5 ⊢ ((sin‘2) · 0) = 0 |
22 | 19, 21 | breqtri 4043 | . . . 4 ⊢ ((sin‘2) · (cos‘2)) < 0 |
23 | 14, 11 | remulcli 8000 | . . . . 5 ⊢ ((sin‘2) · (cos‘2)) ∈ ℝ |
24 | 2pos 9039 | . . . . . 6 ⊢ 0 < 2 | |
25 | 9, 24 | pm3.2i 272 | . . . . 5 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
26 | ltmul2 8842 | . . . . 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 1348 | . . . 4 ⊢ (((sin‘2) · (cos‘2)) < 0 ↔ (2 · ((sin‘2) · (cos‘2))) < (2 · 0)) |
28 | 22, 27 | mpbi 145 | . . 3 ⊢ (2 · ((sin‘2) · (cos‘2))) < (2 · 0) |
29 | 3 | mul01i 8377 | . . 3 ⊢ (2 · 0) = 0 |
30 | 28, 29 | breqtri 4043 | . 2 ⊢ (2 · ((sin‘2) · (cos‘2))) < 0 |
31 | 6, 30 | eqbrtri 4039 | 1 ⊢ (sin‘4) < 0 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 class class class wbr 4018 ‘cfv 5235 (class class class)co 5895 ℂcc 7838 ℝcr 7839 0cc0 7840 · cmul 7845 < clt 8021 2c2 8999 4c4 9001 sincsin 11683 cosccos 11684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 ax-arch 7959 ax-caucvg 7960 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-disj 3996 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-recs 6329 df-irdg 6394 df-frec 6415 df-1o 6440 df-oadd 6444 df-er 6558 df-en 6766 df-dom 6767 df-fin 6768 df-sup 7012 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 df-inn 8949 df-2 9007 df-3 9008 df-4 9009 df-5 9010 df-6 9011 df-7 9012 df-8 9013 df-9 9014 df-n0 9206 df-z 9283 df-uz 9558 df-q 9649 df-rp 9683 df-ioc 9922 df-ico 9923 df-fz 10038 df-fzo 10172 df-seqfrec 10476 df-exp 10550 df-fac 10737 df-bc 10759 df-ihash 10787 df-shft 10855 df-cj 10882 df-re 10883 df-im 10884 df-rsqrt 11038 df-abs 11039 df-clim 11318 df-sumdc 11393 df-ef 11687 df-sin 11689 df-cos 11690 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |