pigt3 - Intuitionistic Logic Explorer

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Theorem pigt3 15868

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 15866	. . . . 5 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))
2	1	simpli 111	. . . 4 ⊢ (sin‘(π / 6)) = (1 / 2)
3		ax-1cn 8262	. . . . 5 ⊢ 1 ∈ ℂ
4		2cn 9354	. . . . . 6 ⊢ 2 ∈ ℂ
5		2ap0 9376	. . . . . 6 ⊢ 2 # 0
6	4, 5	pm3.2i 272	. . . . 5 ⊢ (2 ∈ ℂ ∧ 2 # 0)
7		3cn 9358	. . . . . 6 ⊢ 3 ∈ ℂ
8		3ap0 9379	. . . . . 6 ⊢ 3 # 0
9	7, 8	pm3.2i 272	. . . . 5 ⊢ (3 ∈ ℂ ∧ 3 # 0)
10		divcanap5 9034	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ (3 ∈ ℂ ∧ 3 # 0)) → ((3 · 1) / (3 · 2)) = (1 / 2))
11	3, 6, 9, 10	mp3an 1378	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (1 / 2)
12		3t1e3 9439	. . . . 5 ⊢ (3 · 1) = 3
13		3t2e6 9440	. . . . 5 ⊢ (3 · 2) = 6
14	12, 13	oveq12i 6087	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (3 / 6)
15	2, 11, 14	3eqtr2i 2265	. . 3 ⊢ (sin‘(π / 6)) = (3 / 6)
16		pire 15810	. . . . . . 7 ⊢ π ∈ ℝ
17		6nn 9449	. . . . . . 7 ⊢ 6 ∈ ℕ
18		nndivre 9319	. . . . . . 7 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ)
19	16, 17, 18	mp2an 430	. . . . . 6 ⊢ (π / 6) ∈ ℝ
20		6re 9364	. . . . . . 7 ⊢ 6 ∈ ℝ
21		pipos 15812	. . . . . . 7 ⊢ 0 < π
22		6pos 9384	. . . . . . 7 ⊢ 0 < 6
23	16, 20, 21, 22	divgt0ii 9239	. . . . . 6 ⊢ 0 < (π / 6)
24		1re 8315	. . . . . . 7 ⊢ 1 ∈ ℝ
25		pigt2lt4 15808	. . . . . . . . . 10 ⊢ (2 < π ∧ π < 4)
26	25	simpri 113	. . . . . . . . 9 ⊢ π < 4
27		4re 9360	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
28	16, 27, 20, 22	ltdiv1ii 9249	. . . . . . . . 9 ⊢ (π < 4 ↔ (π / 6) < (4 / 6))
29	26, 28	mpbi 145	. . . . . . . 8 ⊢ (π / 6) < (4 / 6)
30		4lt6 9464	. . . . . . . . 9 ⊢ 4 < 6
31	20, 22	elrpii 10036	. . . . . . . . . 10 ⊢ 6 ∈ ℝ⁺
32		divlt1lt 10104	. . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℝ⁺) → ((4 / 6) < 1 ↔ 4 < 6))
33	27, 31, 32	mp2an 430	. . . . . . . . 9 ⊢ ((4 / 6) < 1 ↔ 4 < 6)
34	30, 33	mpbir 146	. . . . . . . 8 ⊢ (4 / 6) < 1
35		nndivre 9319	. . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℕ) → (4 / 6) ∈ ℝ)
36	27, 17, 35	mp2an 430	. . . . . . . . 9 ⊢ (4 / 6) ∈ ℝ
37	19, 36, 24	lttri 8420	. . . . . . . 8 ⊢ (((π / 6) < (4 / 6) ∧ (4 / 6) < 1) → (π / 6) < 1)
38	29, 34, 37	mp2an 430	. . . . . . 7 ⊢ (π / 6) < 1
39	19, 24, 38	ltleii 8418	. . . . . 6 ⊢ (π / 6) ≤ 1
40		0xr 8362	. . . . . . 7 ⊢ 0 ∈ ℝ^*
41		elioc2 10317	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → ((π / 6) ∈ (0(,]1) ↔ ((π / 6) ∈ ℝ ∧ 0 < (π / 6) ∧ (π / 6) ≤ 1)))
42	40, 24, 41	mp2an 430	. . . . . 6 ⊢ ((π / 6) ∈ (0(,]1) ↔ ((π / 6) ∈ ℝ ∧ 0 < (π / 6) ∧ (π / 6) ≤ 1))
43	19, 23, 39, 42	mpbir3an 1210	. . . . 5 ⊢ (π / 6) ∈ (0(,]1)
44		sin01bnd 12502	. . . . 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 4148	. 2 ⊢ (3 / 6) < (π / 6)
48		3re 9357	. . 3 ⊢ 3 ∈ ℝ
49	48, 16, 20, 22	ltdiv1ii 9249	. 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 1009 = wceq 1402 ∈ wcel 2209 class class class wbr 4125 ‘cfv 5372 (class class class)co 6075 ℂcc 8167 ℝcr 8168 0cc0 8169 1c1 8170 · cmul 8174 ℝ^*cxr 8349 < clt 8350 ≤ cle 8351 − cmin 8487 # cap 8899 / cdiv 8992 ℕcn 9283 2c2 9334 3c3 9335 4c4 9336 6c6 9338 ℝ⁺crp 10033 (,]cioc 10270 ↑cexp 10953 √csqrt 11740 sincsin 12389 cosccos 12390 πcpi 12392

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289 ax-pre-suploc 8290 ax-addf 8291 ax-mulf 8292

This theorem depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-disj 4102 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-isom 5381 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-of 6292 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-frec 6652 df-1o 6677 df-oadd 6681 df-er 6797 df-map 6914 df-pm 6915 df-en 7013 df-dom 7014 df-fin 7015 df-sup 7314 df-inf 7315 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-5 9345 df-6 9346 df-7 9347 df-8 9348 df-9 9349 df-n0 9543 df-z 9624 df-uz 9901 df-q 9999 df-rp 10034 df-xneg 10153 df-xadd 10154 df-ioo 10273 df-ioc 10274 df-ico 10275 df-icc 10276 df-fz 10391 df-fzo 10528 df-seqfrec 10863 df-exp 10954 df-fac 11142 df-bc 11164 df-ihash 11193 df-shft 11558 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-clim 12023 df-sumdc 12098 df-ef 12393 df-sin 12395 df-cos 12396 df-pi 12398 df-rest 13572 df-topgen 13591 df-psmet 14852 df-xmet 14853 df-met 14854 df-bl 14855 df-mopn 14856 df-top 15022 df-topon 15035 df-bases 15067 df-ntr 15120 df-cn 15212 df-cnp 15213 df-tx 15277 df-cncf 15595 df-limced 15680 df-dvap 15681

This theorem is referenced by: pige3 15869

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