pigt3 - Intuitionistic Logic Explorer

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Theorem pigt3 15234

Description: π is greater than 3. (Contributed by Brendan Leahy, 21-Aug-2020.)

**Assertion**
Ref	Expression
pigt3	⊢ 3 < π

**Proof of Theorem pigt3**
Step	Hyp	Ref	Expression
1		sincos6thpi 15232	. . . . 5 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))
2	1	simpli 111	. . . 4 ⊢ (sin‘(π / 6)) = (1 / 2)
3		ax-1cn 8000	. . . . 5 ⊢ 1 ∈ ℂ
4		2cn 9089	. . . . . 6 ⊢ 2 ∈ ℂ
5		2ap0 9111	. . . . . 6 ⊢ 2 # 0
6	4, 5	pm3.2i 272	. . . . 5 ⊢ (2 ∈ ℂ ∧ 2 # 0)
7		3cn 9093	. . . . . 6 ⊢ 3 ∈ ℂ
8		3ap0 9114	. . . . . 6 ⊢ 3 # 0
9	7, 8	pm3.2i 272	. . . . 5 ⊢ (3 ∈ ℂ ∧ 3 # 0)
10		divcanap5 8769	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ (3 ∈ ℂ ∧ 3 # 0)) → ((3 · 1) / (3 · 2)) = (1 / 2))
11	3, 6, 9, 10	mp3an 1349	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (1 / 2)
12		3t1e3 9174	. . . . 5 ⊢ (3 · 1) = 3
13		3t2e6 9175	. . . . 5 ⊢ (3 · 2) = 6
14	12, 13	oveq12i 5946	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (3 / 6)
15	2, 11, 14	3eqtr2i 2231	. . 3 ⊢ (sin‘(π / 6)) = (3 / 6)
16		pire 15176	. . . . . . 7 ⊢ π ∈ ℝ
17		6nn 9184	. . . . . . 7 ⊢ 6 ∈ ℕ
18		nndivre 9054	. . . . . . 7 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ)
19	16, 17, 18	mp2an 426	. . . . . 6 ⊢ (π / 6) ∈ ℝ
20		6re 9099	. . . . . . 7 ⊢ 6 ∈ ℝ
21		pipos 15178	. . . . . . 7 ⊢ 0 < π
22		6pos 9119	. . . . . . 7 ⊢ 0 < 6
23	16, 20, 21, 22	divgt0ii 8974	. . . . . 6 ⊢ 0 < (π / 6)
24		1re 8053	. . . . . . 7 ⊢ 1 ∈ ℝ
25		pigt2lt4 15174	. . . . . . . . . 10 ⊢ (2 < π ∧ π < 4)
26	25	simpri 113	. . . . . . . . 9 ⊢ π < 4
27		4re 9095	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
28	16, 27, 20, 22	ltdiv1ii 8984	. . . . . . . . 9 ⊢ (π < 4 ↔ (π / 6) < (4 / 6))
29	26, 28	mpbi 145	. . . . . . . 8 ⊢ (π / 6) < (4 / 6)
30		4lt6 9199	. . . . . . . . 9 ⊢ 4 < 6
31	20, 22	elrpii 9760	. . . . . . . . . 10 ⊢ 6 ∈ ℝ⁺
32		divlt1lt 9828	. . . . . . . . . 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 9054	. . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℕ) → (4 / 6) ∈ ℝ)
36	27, 17, 35	mp2an 426	. . . . . . . . 9 ⊢ (4 / 6) ∈ ℝ
37	19, 36, 24	lttri 8159	. . . . . . . 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 8157	. . . . . 6 ⊢ (π / 6) ≤ 1
40		0xr 8101	. . . . . . 7 ⊢ 0 ∈ ℝ^*
41		elioc2 10040	. . . . . . 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 1181	. . . . 5 ⊢ (π / 6) ∈ (0(,]1)
44		sin01bnd 11987	. . . . 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 4066	. 2 ⊢ (3 / 6) < (π / 6)
48		3re 9092	. . 3 ⊢ 3 ∈ ℝ
49	48, 16, 20, 22	ltdiv1ii 8984	. 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 980 = wceq 1372 ∈ wcel 2175 class class class wbr 4043 ‘cfv 5268 (class class class)co 5934 ℂcc 7905 ℝcr 7906 0cc0 7907 1c1 7908 · cmul 7912 ℝ^*cxr 8088 < clt 8089 ≤ cle 8090 − cmin 8225 # cap 8636 / cdiv 8727 ℕcn 9018 2c2 9069 3c3 9070 4c4 9071 6c6 9073 ℝ⁺crp 9757 (,]cioc 9993 ↑cexp 10664 √csqrt 11226 sincsin 11874 cosccos 11875 πcpi 11877

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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-iinf 4634 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-mulrcl 8006 ax-addcom 8007 ax-mulcom 8008 ax-addass 8009 ax-mulass 8010 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-1rid 8014 ax-0id 8015 ax-rnegex 8016 ax-precex 8017 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-apti 8022 ax-pre-ltadd 8023 ax-pre-mulgt0 8024 ax-pre-mulext 8025 ax-arch 8026 ax-caucvg 8027 ax-pre-suploc 8028 ax-addf 8029 ax-mulf 8030

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-disj 4021 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4338 df-po 4341 df-iso 4342 df-iord 4411 df-on 4413 df-ilim 4414 df-suc 4416 df-iom 4637 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-res 4685 df-ima 4686 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-f1 5273 df-fo 5274 df-f1o 5275 df-fv 5276 df-isom 5277 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-of 6148 df-1st 6216 df-2nd 6217 df-recs 6381 df-irdg 6446 df-frec 6467 df-1o 6492 df-oadd 6496 df-er 6610 df-map 6727 df-pm 6728 df-en 6818 df-dom 6819 df-fin 6820 df-sup 7068 df-inf 7069 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-reap 8630 df-ap 8637 df-div 8728 df-inn 9019 df-2 9077 df-3 9078 df-4 9079 df-5 9080 df-6 9081 df-7 9082 df-8 9083 df-9 9084 df-n0 9278 df-z 9355 df-uz 9631 df-q 9723 df-rp 9758 df-xneg 9876 df-xadd 9877 df-ioo 9996 df-ioc 9997 df-ico 9998 df-icc 9999 df-fz 10113 df-fzo 10247 df-seqfrec 10574 df-exp 10665 df-fac 10852 df-bc 10874 df-ihash 10902 df-shft 11045 df-cj 11072 df-re 11073 df-im 11074 df-rsqrt 11228 df-abs 11229 df-clim 11509 df-sumdc 11584 df-ef 11878 df-sin 11880 df-cos 11881 df-pi 11883 df-rest 12991 df-topgen 13010 df-psmet 14223 df-xmet 14224 df-met 14225 df-bl 14226 df-mopn 14227 df-top 14388 df-topon 14401 df-bases 14433 df-ntr 14486 df-cn 14578 df-cnp 14579 df-tx 14643 df-cncf 14961 df-limced 15046 df-dvap 15047

This theorem is referenced by: pige3 15235

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