pigt3 - Intuitionistic Logic Explorer

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Theorem pigt3 15567

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 15565	. . . . 5 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))
2	1	simpli 111	. . . 4 ⊢ (sin‘(π / 6)) = (1 / 2)
3		ax-1cn 8124	. . . . 5 ⊢ 1 ∈ ℂ
4		2cn 9213	. . . . . 6 ⊢ 2 ∈ ℂ
5		2ap0 9235	. . . . . 6 ⊢ 2 # 0
6	4, 5	pm3.2i 272	. . . . 5 ⊢ (2 ∈ ℂ ∧ 2 # 0)
7		3cn 9217	. . . . . 6 ⊢ 3 ∈ ℂ
8		3ap0 9238	. . . . . 6 ⊢ 3 # 0
9	7, 8	pm3.2i 272	. . . . 5 ⊢ (3 ∈ ℂ ∧ 3 # 0)
10		divcanap5 8893	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ (3 ∈ ℂ ∧ 3 # 0)) → ((3 · 1) / (3 · 2)) = (1 / 2))
11	3, 6, 9, 10	mp3an 1373	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (1 / 2)
12		3t1e3 9298	. . . . 5 ⊢ (3 · 1) = 3
13		3t2e6 9299	. . . . 5 ⊢ (3 · 2) = 6
14	12, 13	oveq12i 6029	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (3 / 6)
15	2, 11, 14	3eqtr2i 2258	. . 3 ⊢ (sin‘(π / 6)) = (3 / 6)
16		pire 15509	. . . . . . 7 ⊢ π ∈ ℝ
17		6nn 9308	. . . . . . 7 ⊢ 6 ∈ ℕ
18		nndivre 9178	. . . . . . 7 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ)
19	16, 17, 18	mp2an 426	. . . . . 6 ⊢ (π / 6) ∈ ℝ
20		6re 9223	. . . . . . 7 ⊢ 6 ∈ ℝ
21		pipos 15511	. . . . . . 7 ⊢ 0 < π
22		6pos 9243	. . . . . . 7 ⊢ 0 < 6
23	16, 20, 21, 22	divgt0ii 9098	. . . . . 6 ⊢ 0 < (π / 6)
24		1re 8177	. . . . . . 7 ⊢ 1 ∈ ℝ
25		pigt2lt4 15507	. . . . . . . . . 10 ⊢ (2 < π ∧ π < 4)
26	25	simpri 113	. . . . . . . . 9 ⊢ π < 4
27		4re 9219	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
28	16, 27, 20, 22	ltdiv1ii 9108	. . . . . . . . 9 ⊢ (π < 4 ↔ (π / 6) < (4 / 6))
29	26, 28	mpbi 145	. . . . . . . 8 ⊢ (π / 6) < (4 / 6)
30		4lt6 9323	. . . . . . . . 9 ⊢ 4 < 6
31	20, 22	elrpii 9890	. . . . . . . . . 10 ⊢ 6 ∈ ℝ⁺
32		divlt1lt 9958	. . . . . . . . . 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 9178	. . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℕ) → (4 / 6) ∈ ℝ)
36	27, 17, 35	mp2an 426	. . . . . . . . 9 ⊢ (4 / 6) ∈ ℝ
37	19, 36, 24	lttri 8283	. . . . . . . 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 8281	. . . . . 6 ⊢ (π / 6) ≤ 1
40		0xr 8225	. . . . . . 7 ⊢ 0 ∈ ℝ^*
41		elioc2 10170	. . . . . . 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 1205	. . . . 5 ⊢ (π / 6) ∈ (0(,]1)
44		sin01bnd 12317	. . . . 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 4111	. 2 ⊢ (3 / 6) < (π / 6)
48		3re 9216	. . 3 ⊢ 3 ∈ ℝ
49	48, 16, 20, 22	ltdiv1ii 9108	. 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 1004 = wceq 1397 ∈ wcel 2202 class class class wbr 4088 ‘cfv 5326 (class class class)co 6017 ℂcc 8029 ℝcr 8030 0cc0 8031 1c1 8032 · cmul 8036 ℝ^*cxr 8212 < clt 8213 ≤ cle 8214 − cmin 8349 # cap 8760 / cdiv 8851 ℕcn 9142 2c2 9193 3c3 9194 4c4 9195 6c6 9197 ℝ⁺crp 9887 (,]cioc 10123 ↑cexp 10799 √csqrt 11556 sincsin 12204 cosccos 12205 πcpi 12207

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151 ax-pre-suploc 8152 ax-addf 8153 ax-mulf 8154

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-disj 4065 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-of 6234 df-1st 6302 df-2nd 6303 df-recs 6470 df-irdg 6535 df-frec 6556 df-1o 6581 df-oadd 6585 df-er 6701 df-map 6818 df-pm 6819 df-en 6909 df-dom 6910 df-fin 6911 df-sup 7182 df-inf 7183 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-5 9204 df-6 9205 df-7 9206 df-8 9207 df-9 9208 df-n0 9402 df-z 9479 df-uz 9755 df-q 9853 df-rp 9888 df-xneg 10006 df-xadd 10007 df-ioo 10126 df-ioc 10127 df-ico 10128 df-icc 10129 df-fz 10243 df-fzo 10377 df-seqfrec 10709 df-exp 10800 df-fac 10987 df-bc 11009 df-ihash 11037 df-shft 11375 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559 df-clim 11839 df-sumdc 11914 df-ef 12208 df-sin 12210 df-cos 12211 df-pi 12213 df-rest 13323 df-topgen 13342 df-psmet 14556 df-xmet 14557 df-met 14558 df-bl 14559 df-mopn 14560 df-top 14721 df-topon 14734 df-bases 14766 df-ntr 14819 df-cn 14911 df-cnp 14912 df-tx 14976 df-cncf 15294 df-limced 15379 df-dvap 15380

This theorem is referenced by: pige3 15568

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