Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cosz12 | GIF version |
Description: Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and Jim Kingdon, 7-Mar-2024.) |
Ref | Expression |
---|---|
cosz12 | ⊢ ∃𝑝 ∈ (1(,)2)(cos‘𝑝) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 7894 | . . 3 ⊢ (⊤ → 1 ∈ ℝ) | |
2 | 2re 8904 | . . . 4 ⊢ 2 ∈ ℝ | |
3 | 2 | a1i 9 | . . 3 ⊢ (⊤ → 2 ∈ ℝ) |
4 | 0red 7880 | . . 3 ⊢ (⊤ → 0 ∈ ℝ) | |
5 | 1lt2 9003 | . . . 4 ⊢ 1 < 2 | |
6 | 5 | a1i 9 | . . 3 ⊢ (⊤ → 1 < 2) |
7 | 1re 7878 | . . . . . 6 ⊢ 1 ∈ ℝ | |
8 | iccssre 9860 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ) → (1[,]2) ⊆ ℝ) | |
9 | 7, 2, 8 | mp2an 423 | . . . . 5 ⊢ (1[,]2) ⊆ ℝ |
10 | ax-resscn 7825 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
11 | 9, 10 | sstri 3137 | . . . 4 ⊢ (1[,]2) ⊆ ℂ |
12 | 11 | a1i 9 | . . 3 ⊢ (⊤ → (1[,]2) ⊆ ℂ) |
13 | coscn 13133 | . . . 4 ⊢ cos ∈ (ℂ–cn→ℂ) | |
14 | 13 | a1i 9 | . . 3 ⊢ (⊤ → cos ∈ (ℂ–cn→ℂ)) |
15 | 9 | sseli 3124 | . . . . 5 ⊢ (𝑥 ∈ (1[,]2) → 𝑥 ∈ ℝ) |
16 | 15 | recoscld 11625 | . . . 4 ⊢ (𝑥 ∈ (1[,]2) → (cos‘𝑥) ∈ ℝ) |
17 | 16 | adantl 275 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,]2)) → (cos‘𝑥) ∈ ℝ) |
18 | sincos2sgn 11666 | . . . . . 6 ⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) | |
19 | 18 | simpri 112 | . . . . 5 ⊢ (cos‘2) < 0 |
20 | sincos1sgn 11665 | . . . . . 6 ⊢ (0 < (sin‘1) ∧ 0 < (cos‘1)) | |
21 | 20 | simpri 112 | . . . . 5 ⊢ 0 < (cos‘1) |
22 | 19, 21 | pm3.2i 270 | . . . 4 ⊢ ((cos‘2) < 0 ∧ 0 < (cos‘1)) |
23 | 22 | a1i 9 | . . 3 ⊢ (⊤ → ((cos‘2) < 0 ∧ 0 < (cos‘1))) |
24 | cos12dec 11668 | . . . . 5 ⊢ ((𝑥 ∈ (1[,]2) ∧ 𝑦 ∈ (1[,]2) ∧ 𝑥 < 𝑦) → (cos‘𝑦) < (cos‘𝑥)) | |
25 | 24 | 3expb 1186 | . . . 4 ⊢ ((𝑥 ∈ (1[,]2) ∧ (𝑦 ∈ (1[,]2) ∧ 𝑥 < 𝑦)) → (cos‘𝑦) < (cos‘𝑥)) |
26 | 25 | adantll 468 | . . 3 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,]2)) ∧ (𝑦 ∈ (1[,]2) ∧ 𝑥 < 𝑦)) → (cos‘𝑦) < (cos‘𝑥)) |
27 | 1, 3, 4, 6, 12, 14, 17, 23, 26 | ivthdec 13064 | . 2 ⊢ (⊤ → ∃𝑝 ∈ (1(,)2)(cos‘𝑝) = 0) |
28 | 27 | mptru 1344 | 1 ⊢ ∃𝑝 ∈ (1(,)2)(cos‘𝑝) = 0 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1335 ⊤wtru 1336 ∈ wcel 2128 ∃wrex 2436 ⊆ wss 3102 class class class wbr 3966 ‘cfv 5171 (class class class)co 5825 ℂcc 7731 ℝcr 7732 0cc0 7733 1c1 7734 < clt 7913 2c2 8885 (,)cioo 9793 [,]cicc 9796 sincsin 11545 cosccos 11546 –cn→ccncf 12999 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 ax-arch 7852 ax-caucvg 7853 ax-pre-suploc 7854 ax-addf 7855 ax-mulf 7856 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-disj 3944 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-isom 5180 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-of 6033 df-1st 6089 df-2nd 6090 df-recs 6253 df-irdg 6318 df-frec 6339 df-1o 6364 df-oadd 6368 df-er 6481 df-map 6596 df-pm 6597 df-en 6687 df-dom 6688 df-fin 6689 df-sup 6929 df-inf 6930 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 df-inn 8835 df-2 8893 df-3 8894 df-4 8895 df-5 8896 df-6 8897 df-7 8898 df-8 8899 df-9 8900 df-n0 9092 df-z 9169 df-uz 9441 df-q 9530 df-rp 9562 df-xneg 9680 df-xadd 9681 df-ioo 9797 df-ioc 9798 df-ico 9799 df-icc 9800 df-fz 9914 df-fzo 10046 df-seqfrec 10349 df-exp 10423 df-fac 10604 df-bc 10626 df-ihash 10654 df-shft 10719 df-cj 10746 df-re 10747 df-im 10748 df-rsqrt 10902 df-abs 10903 df-clim 11180 df-sumdc 11255 df-ef 11549 df-sin 11551 df-cos 11552 df-rest 12395 df-topgen 12414 df-psmet 12429 df-xmet 12430 df-met 12431 df-bl 12432 df-mopn 12433 df-top 12438 df-topon 12451 df-bases 12483 df-ntr 12538 df-cn 12630 df-cnp 12631 df-tx 12695 df-cncf 13000 df-limced 13067 df-dvap 13068 |
This theorem is referenced by: sin0pilem1 13144 |
Copyright terms: Public domain | W3C validator |