pigt3 - Intuitionistic Logic Explorer

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Theorem pigt3 15391

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 15389	. . . . 5 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))
2	1	simpli 111	. . . 4 ⊢ (sin‘(π / 6)) = (1 / 2)
3		ax-1cn 8038	. . . . 5 ⊢ 1 ∈ ℂ
4		2cn 9127	. . . . . 6 ⊢ 2 ∈ ℂ
5		2ap0 9149	. . . . . 6 ⊢ 2 # 0
6	4, 5	pm3.2i 272	. . . . 5 ⊢ (2 ∈ ℂ ∧ 2 # 0)
7		3cn 9131	. . . . . 6 ⊢ 3 ∈ ℂ
8		3ap0 9152	. . . . . 6 ⊢ 3 # 0
9	7, 8	pm3.2i 272	. . . . 5 ⊢ (3 ∈ ℂ ∧ 3 # 0)
10		divcanap5 8807	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ (3 ∈ ℂ ∧ 3 # 0)) → ((3 · 1) / (3 · 2)) = (1 / 2))
11	3, 6, 9, 10	mp3an 1350	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (1 / 2)
12		3t1e3 9212	. . . . 5 ⊢ (3 · 1) = 3
13		3t2e6 9213	. . . . 5 ⊢ (3 · 2) = 6
14	12, 13	oveq12i 5969	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (3 / 6)
15	2, 11, 14	3eqtr2i 2233	. . 3 ⊢ (sin‘(π / 6)) = (3 / 6)
16		pire 15333	. . . . . . 7 ⊢ π ∈ ℝ
17		6nn 9222	. . . . . . 7 ⊢ 6 ∈ ℕ
18		nndivre 9092	. . . . . . 7 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ)
19	16, 17, 18	mp2an 426	. . . . . 6 ⊢ (π / 6) ∈ ℝ
20		6re 9137	. . . . . . 7 ⊢ 6 ∈ ℝ
21		pipos 15335	. . . . . . 7 ⊢ 0 < π
22		6pos 9157	. . . . . . 7 ⊢ 0 < 6
23	16, 20, 21, 22	divgt0ii 9012	. . . . . 6 ⊢ 0 < (π / 6)
24		1re 8091	. . . . . . 7 ⊢ 1 ∈ ℝ
25		pigt2lt4 15331	. . . . . . . . . 10 ⊢ (2 < π ∧ π < 4)
26	25	simpri 113	. . . . . . . . 9 ⊢ π < 4
27		4re 9133	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
28	16, 27, 20, 22	ltdiv1ii 9022	. . . . . . . . 9 ⊢ (π < 4 ↔ (π / 6) < (4 / 6))
29	26, 28	mpbi 145	. . . . . . . 8 ⊢ (π / 6) < (4 / 6)
30		4lt6 9237	. . . . . . . . 9 ⊢ 4 < 6
31	20, 22	elrpii 9798	. . . . . . . . . 10 ⊢ 6 ∈ ℝ⁺
32		divlt1lt 9866	. . . . . . . . . 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 9092	. . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℕ) → (4 / 6) ∈ ℝ)
36	27, 17, 35	mp2an 426	. . . . . . . . 9 ⊢ (4 / 6) ∈ ℝ
37	19, 36, 24	lttri 8197	. . . . . . . 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 8195	. . . . . 6 ⊢ (π / 6) ≤ 1
40		0xr 8139	. . . . . . 7 ⊢ 0 ∈ ℝ^*
41		elioc2 10078	. . . . . . 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 1182	. . . . 5 ⊢ (π / 6) ∈ (0(,]1)
44		sin01bnd 12143	. . . . 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 4074	. 2 ⊢ (3 / 6) < (π / 6)
48		3re 9130	. . 3 ⊢ 3 ∈ ℝ
49	48, 16, 20, 22	ltdiv1ii 9022	. 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 981 = wceq 1373 ∈ wcel 2177 class class class wbr 4051 ‘cfv 5280 (class class class)co 5957 ℂcc 7943 ℝcr 7944 0cc0 7945 1c1 7946 · cmul 7950 ℝ^*cxr 8126 < clt 8127 ≤ cle 8128 − cmin 8263 # cap 8674 / cdiv 8765 ℕcn 9056 2c2 9107 3c3 9108 4c4 9109 6c6 9111 ℝ⁺crp 9795 (,]cioc 10031 ↑cexp 10705 √csqrt 11382 sincsin 12030 cosccos 12031 πcpi 12033

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4167 ax-sep 4170 ax-nul 4178 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-iinf 4644 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-mulrcl 8044 ax-addcom 8045 ax-mulcom 8046 ax-addass 8047 ax-mulass 8048 ax-distr 8049 ax-i2m1 8050 ax-0lt1 8051 ax-1rid 8052 ax-0id 8053 ax-rnegex 8054 ax-precex 8055 ax-cnre 8056 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061 ax-pre-mulgt0 8062 ax-pre-mulext 8063 ax-arch 8064 ax-caucvg 8065 ax-pre-suploc 8066 ax-addf 8067 ax-mulf 8068

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-if 3576 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-int 3892 df-iun 3935 df-disj 4028 df-br 4052 df-opab 4114 df-mpt 4115 df-tr 4151 df-id 4348 df-po 4351 df-iso 4352 df-iord 4421 df-on 4423 df-ilim 4424 df-suc 4426 df-iom 4647 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-rn 4694 df-res 4695 df-ima 4696 df-iota 5241 df-fun 5282 df-fn 5283 df-f 5284 df-f1 5285 df-fo 5286 df-f1o 5287 df-fv 5288 df-isom 5289 df-riota 5912 df-ov 5960 df-oprab 5961 df-mpo 5962 df-of 6171 df-1st 6239 df-2nd 6240 df-recs 6404 df-irdg 6469 df-frec 6490 df-1o 6515 df-oadd 6519 df-er 6633 df-map 6750 df-pm 6751 df-en 6841 df-dom 6842 df-fin 6843 df-sup 7101 df-inf 7102 df-pnf 8129 df-mnf 8130 df-xr 8131 df-ltxr 8132 df-le 8133 df-sub 8265 df-neg 8266 df-reap 8668 df-ap 8675 df-div 8766 df-inn 9057 df-2 9115 df-3 9116 df-4 9117 df-5 9118 df-6 9119 df-7 9120 df-8 9121 df-9 9122 df-n0 9316 df-z 9393 df-uz 9669 df-q 9761 df-rp 9796 df-xneg 9914 df-xadd 9915 df-ioo 10034 df-ioc 10035 df-ico 10036 df-icc 10037 df-fz 10151 df-fzo 10285 df-seqfrec 10615 df-exp 10706 df-fac 10893 df-bc 10915 df-ihash 10943 df-shft 11201 df-cj 11228 df-re 11229 df-im 11230 df-rsqrt 11384 df-abs 11385 df-clim 11665 df-sumdc 11740 df-ef 12034 df-sin 12036 df-cos 12037 df-pi 12039 df-rest 13148 df-topgen 13167 df-psmet 14380 df-xmet 14381 df-met 14382 df-bl 14383 df-mopn 14384 df-top 14545 df-topon 14558 df-bases 14590 df-ntr 14643 df-cn 14735 df-cnp 14736 df-tx 14800 df-cncf 15118 df-limced 15203 df-dvap 15204

This theorem is referenced by: pige3 15392

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