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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 14199 | . . . . 5 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2)) | |
2 | 1 | simpli 111 | . . . 4 ⊢ (sin‘(π / 6)) = (1 / 2) |
3 | ax-1cn 7903 | . . . . 5 ⊢ 1 ∈ ℂ | |
4 | 2cn 8989 | . . . . . 6 ⊢ 2 ∈ ℂ | |
5 | 2ap0 9011 | . . . . . 6 ⊢ 2 # 0 | |
6 | 4, 5 | pm3.2i 272 | . . . . 5 ⊢ (2 ∈ ℂ ∧ 2 # 0) |
7 | 3cn 8993 | . . . . . 6 ⊢ 3 ∈ ℂ | |
8 | 3ap0 9014 | . . . . . 6 ⊢ 3 # 0 | |
9 | 7, 8 | pm3.2i 272 | . . . . 5 ⊢ (3 ∈ ℂ ∧ 3 # 0) |
10 | divcanap5 8670 | . . . . 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 9073 | . . . . 5 ⊢ (3 · 1) = 3 | |
13 | 3t2e6 9074 | . . . . 5 ⊢ (3 · 2) = 6 | |
14 | 12, 13 | oveq12i 5886 | . . . 4 ⊢ ((3 · 1) / (3 · 2)) = (3 / 6) |
15 | 2, 11, 14 | 3eqtr2i 2204 | . . 3 ⊢ (sin‘(π / 6)) = (3 / 6) |
16 | pire 14143 | . . . . . . 7 ⊢ π ∈ ℝ | |
17 | 6nn 9083 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
18 | nndivre 8954 | . . . . . . 7 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ) | |
19 | 16, 17, 18 | mp2an 426 | . . . . . 6 ⊢ (π / 6) ∈ ℝ |
20 | 6re 8999 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
21 | pipos 14145 | . . . . . . 7 ⊢ 0 < π | |
22 | 6pos 9019 | . . . . . . 7 ⊢ 0 < 6 | |
23 | 16, 20, 21, 22 | divgt0ii 8875 | . . . . . 6 ⊢ 0 < (π / 6) |
24 | 1re 7955 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
25 | pigt2lt4 14141 | . . . . . . . . . 10 ⊢ (2 < π ∧ π < 4) | |
26 | 25 | simpri 113 | . . . . . . . . 9 ⊢ π < 4 |
27 | 4re 8995 | . . . . . . . . . 10 ⊢ 4 ∈ ℝ | |
28 | 16, 27, 20, 22 | ltdiv1ii 8885 | . . . . . . . . 9 ⊢ (π < 4 ↔ (π / 6) < (4 / 6)) |
29 | 26, 28 | mpbi 145 | . . . . . . . 8 ⊢ (π / 6) < (4 / 6) |
30 | 4lt6 9098 | . . . . . . . . 9 ⊢ 4 < 6 | |
31 | 20, 22 | elrpii 9655 | . . . . . . . . . 10 ⊢ 6 ∈ ℝ+ |
32 | divlt1lt 9723 | . . . . . . . . . 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 8954 | . . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℕ) → (4 / 6) ∈ ℝ) | |
36 | 27, 17, 35 | mp2an 426 | . . . . . . . . 9 ⊢ (4 / 6) ∈ ℝ |
37 | 19, 36, 24 | lttri 8061 | . . . . . . . 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 8059 | . . . . . 6 ⊢ (π / 6) ≤ 1 |
40 | 0xr 8003 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
41 | elioc2 9935 | . . . . . . 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 11764 | . . . . 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 4026 | . 2 ⊢ (3 / 6) < (π / 6) |
48 | 3re 8992 | . . 3 ⊢ 3 ∈ ℝ | |
49 | 48, 16, 20, 22 | ltdiv1ii 8885 | . 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 4003 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 ℝcr 7809 0cc0 7810 1c1 7811 · cmul 7815 ℝ*cxr 7990 < clt 7991 ≤ cle 7992 − cmin 8127 # cap 8537 / cdiv 8628 ℕcn 8918 2c2 8969 3c3 8970 4c4 8971 6c6 8973 ℝ+crp 9652 (,]cioc 9888 ↑cexp 10518 √csqrt 11004 sincsin 11651 cosccos 11652 πcpi 11654 |
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 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 ax-pre-suploc 7931 ax-addf 7932 ax-mulf 7933 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-disj 3981 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-of 6082 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-frec 6391 df-1o 6416 df-oadd 6420 df-er 6534 df-map 6649 df-pm 6650 df-en 6740 df-dom 6741 df-fin 6742 df-sup 6982 df-inf 6983 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-5 8980 df-6 8981 df-7 8982 df-8 8983 df-9 8984 df-n0 9176 df-z 9253 df-uz 9528 df-q 9619 df-rp 9653 df-xneg 9771 df-xadd 9772 df-ioo 9891 df-ioc 9892 df-ico 9893 df-icc 9894 df-fz 10008 df-fzo 10142 df-seqfrec 10445 df-exp 10519 df-fac 10705 df-bc 10727 df-ihash 10755 df-shft 10823 df-cj 10850 df-re 10851 df-im 10852 df-rsqrt 11006 df-abs 11007 df-clim 11286 df-sumdc 11361 df-ef 11655 df-sin 11657 df-cos 11658 df-pi 11660 df-rest 12689 df-topgen 12708 df-psmet 13383 df-xmet 13384 df-met 13385 df-bl 13386 df-mopn 13387 df-top 13434 df-topon 13447 df-bases 13479 df-ntr 13532 df-cn 13624 df-cnp 13625 df-tx 13689 df-cncf 13994 df-limced 14061 df-dvap 14062 |
This theorem is referenced by: pige3 14202 |
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