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Mirrors > Home > ILE Home > Th. List > cosq23lt0 | GIF version |
Description: The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.) |
Ref | Expression |
---|---|
cosq23lt0 | ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (cos‘𝐴) < 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioore 9886 | . . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 ∈ ℝ) | |
2 | 1 | recnd 7963 | . . 3 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 ∈ ℂ) |
3 | sinhalfpip 13874 | . . 3 ⊢ (𝐴 ∈ ℂ → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴)) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴)) |
5 | halfpire 13846 | . . . . . 6 ⊢ (π / 2) ∈ ℝ | |
6 | 5 | a1i 9 | . . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (π / 2) ∈ ℝ) |
7 | 6, 1 | readdcld 7964 | . . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) ∈ ℝ) |
8 | pidiv2halves 13849 | . . . . 5 ⊢ ((π / 2) + (π / 2)) = π | |
9 | 5 | rexri 7992 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ* |
10 | 3re 8969 | . . . . . . . . . 10 ⊢ 3 ∈ ℝ | |
11 | 10, 5 | remulcli 7949 | . . . . . . . . 9 ⊢ (3 · (π / 2)) ∈ ℝ |
12 | 11 | rexri 7992 | . . . . . . . 8 ⊢ (3 · (π / 2)) ∈ ℝ* |
13 | elioo2 9895 | . . . . . . . 8 ⊢ (((π / 2) ∈ ℝ* ∧ (3 · (π / 2)) ∈ ℝ*) → (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ (π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2))))) | |
14 | 9, 12, 13 | mp2an 426 | . . . . . . 7 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ (π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2)))) |
15 | 14 | simp2bi 1013 | . . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (π / 2) < 𝐴) |
16 | 6, 1, 6, 15 | ltadd2dd 8356 | . . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + (π / 2)) < ((π / 2) + 𝐴)) |
17 | 8, 16 | eqbrtrrid 4036 | . . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → π < ((π / 2) + 𝐴)) |
18 | 11 | a1i 9 | . . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (3 · (π / 2)) ∈ ℝ) |
19 | 14 | simp3bi 1014 | . . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 < (3 · (π / 2))) |
20 | 1, 18, 6, 19 | ltadd2dd 8356 | . . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) < ((π / 2) + (3 · (π / 2)))) |
21 | ax-1cn 7882 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
22 | 3cn 8970 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
23 | 5 | recni 7947 | . . . . . . . 8 ⊢ (π / 2) ∈ ℂ |
24 | 21, 22, 23 | adddiri 7946 | . . . . . . 7 ⊢ ((1 + 3) · (π / 2)) = ((1 · (π / 2)) + (3 · (π / 2))) |
25 | 3p1e4 9030 | . . . . . . . . 9 ⊢ (3 + 1) = 4 | |
26 | 22, 21, 25 | addcomli 8079 | . . . . . . . 8 ⊢ (1 + 3) = 4 |
27 | 26 | oveq1i 5878 | . . . . . . 7 ⊢ ((1 + 3) · (π / 2)) = (4 · (π / 2)) |
28 | 23 | mulid2i 7938 | . . . . . . . 8 ⊢ (1 · (π / 2)) = (π / 2) |
29 | 28 | oveq1i 5878 | . . . . . . 7 ⊢ ((1 · (π / 2)) + (3 · (π / 2))) = ((π / 2) + (3 · (π / 2))) |
30 | 24, 27, 29 | 3eqtr3ri 2207 | . . . . . 6 ⊢ ((π / 2) + (3 · (π / 2))) = (4 · (π / 2)) |
31 | 4cn 8973 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
32 | 2cn 8966 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
33 | 2ap0 8988 | . . . . . . . 8 ⊢ 2 # 0 | |
34 | 32, 33 | pm3.2i 272 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0) |
35 | picn 13841 | . . . . . . 7 ⊢ π ∈ ℂ | |
36 | div32ap 8625 | . . . . . . 7 ⊢ ((4 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ π ∈ ℂ) → ((4 / 2) · π) = (4 · (π / 2))) | |
37 | 31, 34, 35, 36 | mp3an 1337 | . . . . . 6 ⊢ ((4 / 2) · π) = (4 · (π / 2)) |
38 | 4d2e2 9055 | . . . . . . 7 ⊢ (4 / 2) = 2 | |
39 | 38 | oveq1i 5878 | . . . . . 6 ⊢ ((4 / 2) · π) = (2 · π) |
40 | 30, 37, 39 | 3eqtr2i 2204 | . . . . 5 ⊢ ((π / 2) + (3 · (π / 2))) = (2 · π) |
41 | 20, 40 | breqtrdi 4041 | . . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) < (2 · π)) |
42 | pire 13840 | . . . . . 6 ⊢ π ∈ ℝ | |
43 | 42 | rexri 7992 | . . . . 5 ⊢ π ∈ ℝ* |
44 | 2re 8965 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
45 | 44, 42 | remulcli 7949 | . . . . . 6 ⊢ (2 · π) ∈ ℝ |
46 | 45 | rexri 7992 | . . . . 5 ⊢ (2 · π) ∈ ℝ* |
47 | elioo2 9895 | . . . . 5 ⊢ ((π ∈ ℝ* ∧ (2 · π) ∈ ℝ*) → (((π / 2) + 𝐴) ∈ (π(,)(2 · π)) ↔ (((π / 2) + 𝐴) ∈ ℝ ∧ π < ((π / 2) + 𝐴) ∧ ((π / 2) + 𝐴) < (2 · π)))) | |
48 | 43, 46, 47 | mp2an 426 | . . . 4 ⊢ (((π / 2) + 𝐴) ∈ (π(,)(2 · π)) ↔ (((π / 2) + 𝐴) ∈ ℝ ∧ π < ((π / 2) + 𝐴) ∧ ((π / 2) + 𝐴) < (2 · π))) |
49 | 7, 17, 41, 48 | syl3anbrc 1181 | . . 3 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) ∈ (π(,)(2 · π))) |
50 | sinq34lt0t 13885 | . . 3 ⊢ (((π / 2) + 𝐴) ∈ (π(,)(2 · π)) → (sin‘((π / 2) + 𝐴)) < 0) | |
51 | 49, 50 | syl 14 | . 2 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (sin‘((π / 2) + 𝐴)) < 0) |
52 | 4, 51 | eqbrtrrd 4024 | 1 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (cos‘𝐴) < 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 ‘cfv 5211 (class class class)co 5868 ℂcc 7787 ℝcr 7788 0cc0 7789 1c1 7790 + caddc 7792 · cmul 7794 ℝ*cxr 7968 < clt 7969 # cap 8515 / cdiv 8605 2c2 8946 3c3 8947 4c4 8948 (,)cioo 9862 sincsin 11623 cosccos 11624 πcpi 11626 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 ax-arch 7908 ax-caucvg 7909 ax-pre-suploc 7910 ax-addf 7911 ax-mulf 7912 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-disj 3978 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-po 4292 df-iso 4293 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-isom 5220 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-of 6076 df-1st 6134 df-2nd 6135 df-recs 6299 df-irdg 6364 df-frec 6385 df-1o 6410 df-oadd 6414 df-er 6528 df-map 6643 df-pm 6644 df-en 6734 df-dom 6735 df-fin 6736 df-sup 6976 df-inf 6977 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-inn 8896 df-2 8954 df-3 8955 df-4 8956 df-5 8957 df-6 8958 df-7 8959 df-8 8960 df-9 8961 df-n0 9153 df-z 9230 df-uz 9505 df-q 9596 df-rp 9628 df-xneg 9746 df-xadd 9747 df-ioo 9866 df-ioc 9867 df-ico 9868 df-icc 9869 df-fz 9983 df-fzo 10116 df-seqfrec 10419 df-exp 10493 df-fac 10677 df-bc 10699 df-ihash 10727 df-shft 10795 df-cj 10822 df-re 10823 df-im 10824 df-rsqrt 10978 df-abs 10979 df-clim 11258 df-sumdc 11333 df-ef 11627 df-sin 11629 df-cos 11630 df-pi 11632 df-rest 12625 df-topgen 12644 df-psmet 13120 df-xmet 13121 df-met 13122 df-bl 13123 df-mopn 13124 df-top 13129 df-topon 13142 df-bases 13174 df-ntr 13229 df-cn 13321 df-cnp 13322 df-tx 13386 df-cncf 13691 df-limced 13758 df-dvap 13759 |
This theorem is referenced by: coseq0q4123 13888 cos02pilt1 13905 |
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