pigt3 - Intuitionistic Logic Explorer

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Theorem pigt3 15835

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 15833	. . . . 5 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))
2	1	simpli 111	. . . 4 ⊢ (sin‘(π / 6)) = (1 / 2)
3		ax-1cn 8236	. . . . 5 ⊢ 1 ∈ ℂ
4		2cn 9325	. . . . . 6 ⊢ 2 ∈ ℂ
5		2ap0 9347	. . . . . 6 ⊢ 2 # 0
6	4, 5	pm3.2i 272	. . . . 5 ⊢ (2 ∈ ℂ ∧ 2 # 0)
7		3cn 9329	. . . . . 6 ⊢ 3 ∈ ℂ
8		3ap0 9350	. . . . . 6 ⊢ 3 # 0
9	7, 8	pm3.2i 272	. . . . 5 ⊢ (3 ∈ ℂ ∧ 3 # 0)
10		divcanap5 9005	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ (3 ∈ ℂ ∧ 3 # 0)) → ((3 · 1) / (3 · 2)) = (1 / 2))
11	3, 6, 9, 10	mp3an 1374	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (1 / 2)
12		3t1e3 9410	. . . . 5 ⊢ (3 · 1) = 3
13		3t2e6 9411	. . . . 5 ⊢ (3 · 2) = 6
14	12, 13	oveq12i 6070	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (3 / 6)
15	2, 11, 14	3eqtr2i 2261	. . 3 ⊢ (sin‘(π / 6)) = (3 / 6)
16		pire 15777	. . . . . . 7 ⊢ π ∈ ℝ
17		6nn 9420	. . . . . . 7 ⊢ 6 ∈ ℕ
18		nndivre 9290	. . . . . . 7 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ)
19	16, 17, 18	mp2an 426	. . . . . 6 ⊢ (π / 6) ∈ ℝ
20		6re 9335	. . . . . . 7 ⊢ 6 ∈ ℝ
21		pipos 15779	. . . . . . 7 ⊢ 0 < π
22		6pos 9355	. . . . . . 7 ⊢ 0 < 6
23	16, 20, 21, 22	divgt0ii 9210	. . . . . 6 ⊢ 0 < (π / 6)
24		1re 8289	. . . . . . 7 ⊢ 1 ∈ ℝ
25		pigt2lt4 15775	. . . . . . . . . 10 ⊢ (2 < π ∧ π < 4)
26	25	simpri 113	. . . . . . . . 9 ⊢ π < 4
27		4re 9331	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
28	16, 27, 20, 22	ltdiv1ii 9220	. . . . . . . . 9 ⊢ (π < 4 ↔ (π / 6) < (4 / 6))
29	26, 28	mpbi 145	. . . . . . . 8 ⊢ (π / 6) < (4 / 6)
30		4lt6 9435	. . . . . . . . 9 ⊢ 4 < 6
31	20, 22	elrpii 10007	. . . . . . . . . 10 ⊢ 6 ∈ ℝ⁺
32		divlt1lt 10075	. . . . . . . . . 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 9290	. . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℕ) → (4 / 6) ∈ ℝ)
36	27, 17, 35	mp2an 426	. . . . . . . . 9 ⊢ (4 / 6) ∈ ℝ
37	19, 36, 24	lttri 8394	. . . . . . . 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 8392	. . . . . 6 ⊢ (π / 6) ≤ 1
40		0xr 8336	. . . . . . 7 ⊢ 0 ∈ ℝ^*
41		elioc2 10288	. . . . . . 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 1206	. . . . 5 ⊢ (π / 6) ∈ (0(,]1)
44		sin01bnd 12468	. . . . 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 4137	. 2 ⊢ (3 / 6) < (π / 6)
48		3re 9328	. . 3 ⊢ 3 ∈ ℝ
49	48, 16, 20, 22	ltdiv1ii 9220	. 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 1005 = wceq 1398 ∈ wcel 2205 class class class wbr 4114 ‘cfv 5357 (class class class)co 6058 ℂcc 8141 ℝcr 8142 0cc0 8143 1c1 8144 · cmul 8148 ℝ^*cxr 8323 < clt 8324 ≤ cle 8325 − cmin 8460 # cap 8872 / cdiv 8963 ℕcn 9254 2c2 9305 3c3 9306 4c4 9307 6c6 9309 ℝ⁺crp 10004 (,]cioc 10241 ↑cexp 10924 √csqrt 11706 sincsin 12355 cosccos 12356 πcpi 12358

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263 ax-pre-suploc 8264 ax-addf 8265 ax-mulf 8266

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-disj 4091 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-of 6275 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-oadd 6664 df-er 6780 df-map 6897 df-pm 6898 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-5 9316 df-6 9317 df-7 9318 df-8 9319 df-9 9320 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-xneg 10124 df-xadd 10125 df-ioo 10244 df-ioc 10245 df-ico 10246 df-icc 10247 df-fz 10362 df-fzo 10499 df-seqfrec 10834 df-exp 10925 df-fac 11113 df-bc 11135 df-ihash 11164 df-shft 11525 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-clim 11989 df-sumdc 12064 df-ef 12359 df-sin 12361 df-cos 12362 df-pi 12364 df-rest 13538 df-topgen 13557 df-psmet 14817 df-xmet 14818 df-met 14819 df-bl 14820 df-mopn 14821 df-top 14989 df-topon 15002 df-bases 15034 df-ntr 15087 df-cn 15179 df-cnp 15180 df-tx 15244 df-cncf 15562 df-limced 15647 df-dvap 15648

This theorem is referenced by: pige3 15836

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