pigt3 - Intuitionistic Logic Explorer

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Theorem pigt3 15188

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 15186	. . . . 5 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))
2	1	simpli 111	. . . 4 ⊢ (sin‘(π / 6)) = (1 / 2)
3		ax-1cn 7991	. . . . 5 ⊢ 1 ∈ ℂ
4		2cn 9080	. . . . . 6 ⊢ 2 ∈ ℂ
5		2ap0 9102	. . . . . 6 ⊢ 2 # 0
6	4, 5	pm3.2i 272	. . . . 5 ⊢ (2 ∈ ℂ ∧ 2 # 0)
7		3cn 9084	. . . . . 6 ⊢ 3 ∈ ℂ
8		3ap0 9105	. . . . . 6 ⊢ 3 # 0
9	7, 8	pm3.2i 272	. . . . 5 ⊢ (3 ∈ ℂ ∧ 3 # 0)
10		divcanap5 8760	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ (3 ∈ ℂ ∧ 3 # 0)) → ((3 · 1) / (3 · 2)) = (1 / 2))
11	3, 6, 9, 10	mp3an 1348	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (1 / 2)
12		3t1e3 9165	. . . . 5 ⊢ (3 · 1) = 3
13		3t2e6 9166	. . . . 5 ⊢ (3 · 2) = 6
14	12, 13	oveq12i 5937	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (3 / 6)
15	2, 11, 14	3eqtr2i 2223	. . 3 ⊢ (sin‘(π / 6)) = (3 / 6)
16		pire 15130	. . . . . . 7 ⊢ π ∈ ℝ
17		6nn 9175	. . . . . . 7 ⊢ 6 ∈ ℕ
18		nndivre 9045	. . . . . . 7 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ)
19	16, 17, 18	mp2an 426	. . . . . 6 ⊢ (π / 6) ∈ ℝ
20		6re 9090	. . . . . . 7 ⊢ 6 ∈ ℝ
21		pipos 15132	. . . . . . 7 ⊢ 0 < π
22		6pos 9110	. . . . . . 7 ⊢ 0 < 6
23	16, 20, 21, 22	divgt0ii 8965	. . . . . 6 ⊢ 0 < (π / 6)
24		1re 8044	. . . . . . 7 ⊢ 1 ∈ ℝ
25		pigt2lt4 15128	. . . . . . . . . 10 ⊢ (2 < π ∧ π < 4)
26	25	simpri 113	. . . . . . . . 9 ⊢ π < 4
27		4re 9086	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
28	16, 27, 20, 22	ltdiv1ii 8975	. . . . . . . . 9 ⊢ (π < 4 ↔ (π / 6) < (4 / 6))
29	26, 28	mpbi 145	. . . . . . . 8 ⊢ (π / 6) < (4 / 6)
30		4lt6 9190	. . . . . . . . 9 ⊢ 4 < 6
31	20, 22	elrpii 9750	. . . . . . . . . 10 ⊢ 6 ∈ ℝ⁺
32		divlt1lt 9818	. . . . . . . . . 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 9045	. . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℕ) → (4 / 6) ∈ ℝ)
36	27, 17, 35	mp2an 426	. . . . . . . . 9 ⊢ (4 / 6) ∈ ℝ
37	19, 36, 24	lttri 8150	. . . . . . . 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 8148	. . . . . 6 ⊢ (π / 6) ≤ 1
40		0xr 8092	. . . . . . 7 ⊢ 0 ∈ ℝ^*
41		elioc2 10030	. . . . . . 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 1181	. . . . 5 ⊢ (π / 6) ∈ (0(,]1)
44		sin01bnd 11941	. . . . 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 4057	. 2 ⊢ (3 / 6) < (π / 6)
48		3re 9083	. . 3 ⊢ 3 ∈ ℝ
49	48, 16, 20, 22	ltdiv1ii 8975	. 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 980 = wceq 1364 ∈ wcel 2167 class class class wbr 4034 ‘cfv 5259 (class class class)co 5925 ℂcc 7896 ℝcr 7897 0cc0 7898 1c1 7899 · cmul 7903 ℝ^*cxr 8079 < clt 8080 ≤ cle 8081 − cmin 8216 # cap 8627 / cdiv 8718 ℕcn 9009 2c2 9060 3c3 9061 4c4 9062 6c6 9064 ℝ⁺crp 9747 (,]cioc 9983 ↑cexp 10649 √csqrt 11180 sincsin 11828 cosccos 11829 πcpi 11831

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7989 ax-resscn 7990 ax-1cn 7991 ax-1re 7992 ax-icn 7993 ax-addcl 7994 ax-addrcl 7995 ax-mulcl 7996 ax-mulrcl 7997 ax-addcom 7998 ax-mulcom 7999 ax-addass 8000 ax-mulass 8001 ax-distr 8002 ax-i2m1 8003 ax-0lt1 8004 ax-1rid 8005 ax-0id 8006 ax-rnegex 8007 ax-precex 8008 ax-cnre 8009 ax-pre-ltirr 8010 ax-pre-ltwlin 8011 ax-pre-lttrn 8012 ax-pre-apti 8013 ax-pre-ltadd 8014 ax-pre-mulgt0 8015 ax-pre-mulext 8016 ax-arch 8017 ax-caucvg 8018 ax-pre-suploc 8019 ax-addf 8020 ax-mulf 8021

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-disj 4012 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-of 6139 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-frec 6458 df-1o 6483 df-oadd 6487 df-er 6601 df-map 6718 df-pm 6719 df-en 6809 df-dom 6810 df-fin 6811 df-sup 7059 df-inf 7060 df-pnf 8082 df-mnf 8083 df-xr 8084 df-ltxr 8085 df-le 8086 df-sub 8218 df-neg 8219 df-reap 8621 df-ap 8628 df-div 8719 df-inn 9010 df-2 9068 df-3 9069 df-4 9070 df-5 9071 df-6 9072 df-7 9073 df-8 9074 df-9 9075 df-n0 9269 df-z 9346 df-uz 9621 df-q 9713 df-rp 9748 df-xneg 9866 df-xadd 9867 df-ioo 9986 df-ioc 9987 df-ico 9988 df-icc 9989 df-fz 10103 df-fzo 10237 df-seqfrec 10559 df-exp 10650 df-fac 10837 df-bc 10859 df-ihash 10887 df-shft 10999 df-cj 11026 df-re 11027 df-im 11028 df-rsqrt 11182 df-abs 11183 df-clim 11463 df-sumdc 11538 df-ef 11832 df-sin 11834 df-cos 11835 df-pi 11837 df-rest 12945 df-topgen 12964 df-psmet 14177 df-xmet 14178 df-met 14179 df-bl 14180 df-mopn 14181 df-top 14342 df-topon 14355 df-bases 14387 df-ntr 14440 df-cn 14532 df-cnp 14533 df-tx 14597 df-cncf 14915 df-limced 15000 df-dvap 15001

This theorem is referenced by: pige3 15189

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