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Mirrors > Home > ILE Home > Th. List > pigt3 | GIF version |
Description: π is greater than 3. (Contributed by Brendan Leahy, 21-Aug-2020.) |
Ref | Expression |
---|---|
pigt3 | ⊢ 3 < π |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sincos6thpi 14348 | . . . . 5 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2)) | |
2 | 1 | simpli 111 | . . . 4 ⊢ (sin‘(π / 6)) = (1 / 2) |
3 | ax-1cn 7906 | . . . . 5 ⊢ 1 ∈ ℂ | |
4 | 2cn 8992 | . . . . . 6 ⊢ 2 ∈ ℂ | |
5 | 2ap0 9014 | . . . . . 6 ⊢ 2 # 0 | |
6 | 4, 5 | pm3.2i 272 | . . . . 5 ⊢ (2 ∈ ℂ ∧ 2 # 0) |
7 | 3cn 8996 | . . . . . 6 ⊢ 3 ∈ ℂ | |
8 | 3ap0 9017 | . . . . . 6 ⊢ 3 # 0 | |
9 | 7, 8 | pm3.2i 272 | . . . . 5 ⊢ (3 ∈ ℂ ∧ 3 # 0) |
10 | divcanap5 8673 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ (3 ∈ ℂ ∧ 3 # 0)) → ((3 · 1) / (3 · 2)) = (1 / 2)) | |
11 | 3, 6, 9, 10 | mp3an 1337 | . . . 4 ⊢ ((3 · 1) / (3 · 2)) = (1 / 2) |
12 | 3t1e3 9076 | . . . . 5 ⊢ (3 · 1) = 3 | |
13 | 3t2e6 9077 | . . . . 5 ⊢ (3 · 2) = 6 | |
14 | 12, 13 | oveq12i 5889 | . . . 4 ⊢ ((3 · 1) / (3 · 2)) = (3 / 6) |
15 | 2, 11, 14 | 3eqtr2i 2204 | . . 3 ⊢ (sin‘(π / 6)) = (3 / 6) |
16 | pire 14292 | . . . . . . 7 ⊢ π ∈ ℝ | |
17 | 6nn 9086 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
18 | nndivre 8957 | . . . . . . 7 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ) | |
19 | 16, 17, 18 | mp2an 426 | . . . . . 6 ⊢ (π / 6) ∈ ℝ |
20 | 6re 9002 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
21 | pipos 14294 | . . . . . . 7 ⊢ 0 < π | |
22 | 6pos 9022 | . . . . . . 7 ⊢ 0 < 6 | |
23 | 16, 20, 21, 22 | divgt0ii 8878 | . . . . . 6 ⊢ 0 < (π / 6) |
24 | 1re 7958 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
25 | pigt2lt4 14290 | . . . . . . . . . 10 ⊢ (2 < π ∧ π < 4) | |
26 | 25 | simpri 113 | . . . . . . . . 9 ⊢ π < 4 |
27 | 4re 8998 | . . . . . . . . . 10 ⊢ 4 ∈ ℝ | |
28 | 16, 27, 20, 22 | ltdiv1ii 8888 | . . . . . . . . 9 ⊢ (π < 4 ↔ (π / 6) < (4 / 6)) |
29 | 26, 28 | mpbi 145 | . . . . . . . 8 ⊢ (π / 6) < (4 / 6) |
30 | 4lt6 9101 | . . . . . . . . 9 ⊢ 4 < 6 | |
31 | 20, 22 | elrpii 9658 | . . . . . . . . . 10 ⊢ 6 ∈ ℝ+ |
32 | divlt1lt 9726 | . . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℝ+) → ((4 / 6) < 1 ↔ 4 < 6)) | |
33 | 27, 31, 32 | mp2an 426 | . . . . . . . . 9 ⊢ ((4 / 6) < 1 ↔ 4 < 6) |
34 | 30, 33 | mpbir 146 | . . . . . . . 8 ⊢ (4 / 6) < 1 |
35 | nndivre 8957 | . . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℕ) → (4 / 6) ∈ ℝ) | |
36 | 27, 17, 35 | mp2an 426 | . . . . . . . . 9 ⊢ (4 / 6) ∈ ℝ |
37 | 19, 36, 24 | lttri 8064 | . . . . . . . 8 ⊢ (((π / 6) < (4 / 6) ∧ (4 / 6) < 1) → (π / 6) < 1) |
38 | 29, 34, 37 | mp2an 426 | . . . . . . 7 ⊢ (π / 6) < 1 |
39 | 19, 24, 38 | ltleii 8062 | . . . . . 6 ⊢ (π / 6) ≤ 1 |
40 | 0xr 8006 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
41 | elioc2 9938 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → ((π / 6) ∈ (0(,]1) ↔ ((π / 6) ∈ ℝ ∧ 0 < (π / 6) ∧ (π / 6) ≤ 1))) | |
42 | 40, 24, 41 | mp2an 426 | . . . . . 6 ⊢ ((π / 6) ∈ (0(,]1) ↔ ((π / 6) ∈ ℝ ∧ 0 < (π / 6) ∧ (π / 6) ≤ 1)) |
43 | 19, 23, 39, 42 | mpbir3an 1179 | . . . . 5 ⊢ (π / 6) ∈ (0(,]1) |
44 | sin01bnd 11767 | . . . . 5 ⊢ ((π / 6) ∈ (0(,]1) → (((π / 6) − (((π / 6)↑3) / 3)) < (sin‘(π / 6)) ∧ (sin‘(π / 6)) < (π / 6))) | |
45 | 43, 44 | ax-mp 5 | . . . 4 ⊢ (((π / 6) − (((π / 6)↑3) / 3)) < (sin‘(π / 6)) ∧ (sin‘(π / 6)) < (π / 6)) |
46 | 45 | simpri 113 | . . 3 ⊢ (sin‘(π / 6)) < (π / 6) |
47 | 15, 46 | eqbrtrri 4028 | . 2 ⊢ (3 / 6) < (π / 6) |
48 | 3re 8995 | . . 3 ⊢ 3 ∈ ℝ | |
49 | 48, 16, 20, 22 | ltdiv1ii 8888 | . 2 ⊢ (3 < π ↔ (3 / 6) < (π / 6)) |
50 | 47, 49 | mpbir 146 | 1 ⊢ 3 < π |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 class class class wbr 4005 ‘cfv 5218 (class class class)co 5877 ℂcc 7811 ℝcr 7812 0cc0 7813 1c1 7814 · cmul 7818 ℝ*cxr 7993 < clt 7994 ≤ cle 7995 − cmin 8130 # cap 8540 / cdiv 8631 ℕcn 8921 2c2 8972 3c3 8973 4c4 8974 6c6 8976 ℝ+crp 9655 (,]cioc 9891 ↑cexp 10521 √csqrt 11007 sincsin 11654 cosccos 11655 πcpi 11657 |
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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933 ax-pre-suploc 7934 ax-addf 7935 ax-mulf 7936 |
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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-disj 3983 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-of 6085 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-map 6652 df-pm 6653 df-en 6743 df-dom 6744 df-fin 6745 df-sup 6985 df-inf 6986 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-5 8983 df-6 8984 df-7 8985 df-8 8986 df-9 8987 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-xneg 9774 df-xadd 9775 df-ioo 9894 df-ioc 9895 df-ico 9896 df-icc 9897 df-fz 10011 df-fzo 10145 df-seqfrec 10448 df-exp 10522 df-fac 10708 df-bc 10730 df-ihash 10758 df-shft 10826 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-clim 11289 df-sumdc 11364 df-ef 11658 df-sin 11660 df-cos 11661 df-pi 11663 df-rest 12695 df-topgen 12714 df-psmet 13532 df-xmet 13533 df-met 13534 df-bl 13535 df-mopn 13536 df-top 13583 df-topon 13596 df-bases 13628 df-ntr 13681 df-cn 13773 df-cnp 13774 df-tx 13838 df-cncf 14143 df-limced 14210 df-dvap 14211 |
This theorem is referenced by: pige3 14351 |
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