pigt3 - Intuitionistic Logic Explorer

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Theorem pigt3 13235

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 13233	. . . . 5 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))
2	1	simpli 110	. . . 4 ⊢ (sin‘(π / 6)) = (1 / 2)
3		ax-1cn 7827	. . . . 5 ⊢ 1 ∈ ℂ
4		2cn 8909	. . . . . 6 ⊢ 2 ∈ ℂ
5		2ap0 8931	. . . . . 6 ⊢ 2 # 0
6	4, 5	pm3.2i 270	. . . . 5 ⊢ (2 ∈ ℂ ∧ 2 # 0)
7		3cn 8913	. . . . . 6 ⊢ 3 ∈ ℂ
8		3ap0 8934	. . . . . 6 ⊢ 3 # 0
9	7, 8	pm3.2i 270	. . . . 5 ⊢ (3 ∈ ℂ ∧ 3 # 0)
10		divcanap5 8591	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ (3 ∈ ℂ ∧ 3 # 0)) → ((3 · 1) / (3 · 2)) = (1 / 2))
11	3, 6, 9, 10	mp3an 1319	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (1 / 2)
12		3t1e3 8993	. . . . 5 ⊢ (3 · 1) = 3
13		3t2e6 8994	. . . . 5 ⊢ (3 · 2) = 6
14	12, 13	oveq12i 5838	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (3 / 6)
15	2, 11, 14	3eqtr2i 2184	. . 3 ⊢ (sin‘(π / 6)) = (3 / 6)
16		pire 13177	. . . . . . 7 ⊢ π ∈ ℝ
17		6nn 9003	. . . . . . 7 ⊢ 6 ∈ ℕ
18		nndivre 8874	. . . . . . 7 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ)
19	16, 17, 18	mp2an 423	. . . . . 6 ⊢ (π / 6) ∈ ℝ
20		6re 8919	. . . . . . 7 ⊢ 6 ∈ ℝ
21		pipos 13179	. . . . . . 7 ⊢ 0 < π
22		6pos 8939	. . . . . . 7 ⊢ 0 < 6
23	16, 20, 21, 22	divgt0ii 8795	. . . . . 6 ⊢ 0 < (π / 6)
24		1re 7879	. . . . . . 7 ⊢ 1 ∈ ℝ
25		pigt2lt4 13175	. . . . . . . . . 10 ⊢ (2 < π ∧ π < 4)
26	25	simpri 112	. . . . . . . . 9 ⊢ π < 4
27		4re 8915	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
28	16, 27, 20, 22	ltdiv1ii 8805	. . . . . . . . 9 ⊢ (π < 4 ↔ (π / 6) < (4 / 6))
29	26, 28	mpbi 144	. . . . . . . 8 ⊢ (π / 6) < (4 / 6)
30		4lt6 9018	. . . . . . . . 9 ⊢ 4 < 6
31	20, 22	elrpii 9569	. . . . . . . . . 10 ⊢ 6 ∈ ℝ⁺
32		divlt1lt 9637	. . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℝ⁺) → ((4 / 6) < 1 ↔ 4 < 6))
33	27, 31, 32	mp2an 423	. . . . . . . . 9 ⊢ ((4 / 6) < 1 ↔ 4 < 6)
34	30, 33	mpbir 145	. . . . . . . 8 ⊢ (4 / 6) < 1
35		nndivre 8874	. . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℕ) → (4 / 6) ∈ ℝ)
36	27, 17, 35	mp2an 423	. . . . . . . . 9 ⊢ (4 / 6) ∈ ℝ
37	19, 36, 24	lttri 7984	. . . . . . . 8 ⊢ (((π / 6) < (4 / 6) ∧ (4 / 6) < 1) → (π / 6) < 1)
38	29, 34, 37	mp2an 423	. . . . . . 7 ⊢ (π / 6) < 1
39	19, 24, 38	ltleii 7982	. . . . . 6 ⊢ (π / 6) ≤ 1
40		0xr 7926	. . . . . . 7 ⊢ 0 ∈ ℝ^*
41		elioc2 9846	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → ((π / 6) ∈ (0(,]1) ↔ ((π / 6) ∈ ℝ ∧ 0 < (π / 6) ∧ (π / 6) ≤ 1)))
42	40, 24, 41	mp2an 423	. . . . . 6 ⊢ ((π / 6) ∈ (0(,]1) ↔ ((π / 6) ∈ ℝ ∧ 0 < (π / 6) ∧ (π / 6) ≤ 1))
43	19, 23, 39, 42	mpbir3an 1164	. . . . 5 ⊢ (π / 6) ∈ (0(,]1)
44		sin01bnd 11665	. . . . 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 112	. . 3 ⊢ (sin‘(π / 6)) < (π / 6)
47	15, 46	eqbrtrri 3989	. 2 ⊢ (3 / 6) < (π / 6)
48		3re 8912	. . 3 ⊢ 3 ∈ ℝ
49	48, 16, 20, 22	ltdiv1ii 8805	. 2 ⊢ (3 < π ↔ (3 / 6) < (π / 6))
50	47, 49	mpbir 145	1 ⊢ 3 < π

Colors of variables: wff set class

Syntax hints: ∧ wa 103 ↔ wb 104 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 class class class wbr 3967 ‘cfv 5172 (class class class)co 5826 ℂcc 7732 ℝcr 7733 0cc0 7734 1c1 7735 · cmul 7739 ℝ^*cxr 7913 < clt 7914 ≤ cle 7915 − cmin 8050 # cap 8460 / cdiv 8549 ℕcn 8838 2c2 8889 3c3 8890 4c4 8891 6c6 8893 ℝ⁺crp 9566 (,]cioc 9799 ↑cexp 10427 √csqrt 10907 sincsin 11552 cosccos 11553 πcpi 11555

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 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 ax-pre-mulext 7852 ax-arch 7853 ax-caucvg 7854 ax-pre-suploc 7855 ax-addf 7856 ax-mulf 7857

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 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-disj 3945 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-id 4255 df-po 4258 df-iso 4259 df-iord 4328 df-on 4330 df-ilim 4331 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-isom 5181 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-of 6034 df-1st 6090 df-2nd 6091 df-recs 6254 df-irdg 6319 df-frec 6340 df-1o 6365 df-oadd 6369 df-er 6482 df-map 6597 df-pm 6598 df-en 6688 df-dom 6689 df-fin 6690 df-sup 6930 df-inf 6931 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 df-div 8550 df-inn 8839 df-2 8897 df-3 8898 df-4 8899 df-5 8900 df-6 8901 df-7 8902 df-8 8903 df-9 8904 df-n0 9096 df-z 9173 df-uz 9445 df-q 9535 df-rp 9567 df-xneg 9685 df-xadd 9686 df-ioo 9802 df-ioc 9803 df-ico 9804 df-icc 9805 df-fz 9919 df-fzo 10051 df-seqfrec 10354 df-exp 10428 df-fac 10611 df-bc 10633 df-ihash 10661 df-shft 10726 df-cj 10753 df-re 10754 df-im 10755 df-rsqrt 10909 df-abs 10910 df-clim 11187 df-sumdc 11262 df-ef 11556 df-sin 11558 df-cos 11559 df-pi 11561 df-rest 12423 df-topgen 12442 df-psmet 12457 df-xmet 12458 df-met 12459 df-bl 12460 df-mopn 12461 df-top 12466 df-topon 12479 df-bases 12511 df-ntr 12566 df-cn 12658 df-cnp 12659 df-tx 12723 df-cncf 13028 df-limced 13095 df-dvap 13096

This theorem is referenced by: pige3 13236

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