pigt3 - Intuitionistic Logic Explorer

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Theorem pigt3 12973

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 12971	. . . . 5 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))
2	1	simpli 110	. . . 4 ⊢ (sin‘(π / 6)) = (1 / 2)
3		ax-1cn 7737	. . . . 5 ⊢ 1 ∈ ℂ
4		2cn 8815	. . . . . 6 ⊢ 2 ∈ ℂ
5		2ap0 8837	. . . . . 6 ⊢ 2 # 0
6	4, 5	pm3.2i 270	. . . . 5 ⊢ (2 ∈ ℂ ∧ 2 # 0)
7		3cn 8819	. . . . . 6 ⊢ 3 ∈ ℂ
8		3ap0 8840	. . . . . 6 ⊢ 3 # 0
9	7, 8	pm3.2i 270	. . . . 5 ⊢ (3 ∈ ℂ ∧ 3 # 0)
10		divcanap5 8498	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ (3 ∈ ℂ ∧ 3 # 0)) → ((3 · 1) / (3 · 2)) = (1 / 2))
11	3, 6, 9, 10	mp3an 1316	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (1 / 2)
12		3t1e3 8899	. . . . 5 ⊢ (3 · 1) = 3
13		3t2e6 8900	. . . . 5 ⊢ (3 · 2) = 6
14	12, 13	oveq12i 5794	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (3 / 6)
15	2, 11, 14	3eqtr2i 2167	. . 3 ⊢ (sin‘(π / 6)) = (3 / 6)
16		pire 12915	. . . . . . 7 ⊢ π ∈ ℝ
17		6nn 8909	. . . . . . 7 ⊢ 6 ∈ ℕ
18		nndivre 8780	. . . . . . 7 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ)
19	16, 17, 18	mp2an 423	. . . . . 6 ⊢ (π / 6) ∈ ℝ
20		6re 8825	. . . . . . 7 ⊢ 6 ∈ ℝ
21		pipos 12917	. . . . . . 7 ⊢ 0 < π
22		6pos 8845	. . . . . . 7 ⊢ 0 < 6
23	16, 20, 21, 22	divgt0ii 8701	. . . . . 6 ⊢ 0 < (π / 6)
24		1re 7789	. . . . . . 7 ⊢ 1 ∈ ℝ
25		pigt2lt4 12913	. . . . . . . . . 10 ⊢ (2 < π ∧ π < 4)
26	25	simpri 112	. . . . . . . . 9 ⊢ π < 4
27		4re 8821	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
28	16, 27, 20, 22	ltdiv1ii 8711	. . . . . . . . 9 ⊢ (π < 4 ↔ (π / 6) < (4 / 6))
29	26, 28	mpbi 144	. . . . . . . 8 ⊢ (π / 6) < (4 / 6)
30		4lt6 8924	. . . . . . . . 9 ⊢ 4 < 6
31	20, 22	elrpii 9473	. . . . . . . . . 10 ⊢ 6 ∈ ℝ⁺
32		divlt1lt 9541	. . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℝ⁺) → ((4 / 6) < 1 ↔ 4 < 6))
33	27, 31, 32	mp2an 423	. . . . . . . . 9 ⊢ ((4 / 6) < 1 ↔ 4 < 6)
34	30, 33	mpbir 145	. . . . . . . 8 ⊢ (4 / 6) < 1
35		nndivre 8780	. . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℕ) → (4 / 6) ∈ ℝ)
36	27, 17, 35	mp2an 423	. . . . . . . . 9 ⊢ (4 / 6) ∈ ℝ
37	19, 36, 24	lttri 7892	. . . . . . . 8 ⊢ (((π / 6) < (4 / 6) ∧ (4 / 6) < 1) → (π / 6) < 1)
38	29, 34, 37	mp2an 423	. . . . . . 7 ⊢ (π / 6) < 1
39	19, 24, 38	ltleii 7890	. . . . . 6 ⊢ (π / 6) ≤ 1
40		0xr 7836	. . . . . . 7 ⊢ 0 ∈ ℝ^*
41		elioc2 9749	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → ((π / 6) ∈ (0(,]1) ↔ ((π / 6) ∈ ℝ ∧ 0 < (π / 6) ∧ (π / 6) ≤ 1)))
42	40, 24, 41	mp2an 423	. . . . . 6 ⊢ ((π / 6) ∈ (0(,]1) ↔ ((π / 6) ∈ ℝ ∧ 0 < (π / 6) ∧ (π / 6) ≤ 1))
43	19, 23, 39, 42	mpbir3an 1164	. . . . 5 ⊢ (π / 6) ∈ (0(,]1)
44		sin01bnd 11500	. . . . 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 112	. . 3 ⊢ (sin‘(π / 6)) < (π / 6)
47	15, 46	eqbrtrri 3959	. 2 ⊢ (3 / 6) < (π / 6)
48		3re 8818	. . 3 ⊢ 3 ∈ ℝ
49	48, 16, 20, 22	ltdiv1ii 8711	. 2 ⊢ (3 < π ↔ (3 / 6) < (π / 6))
50	47, 49	mpbir 145	1 ⊢ 3 < π

Colors of variables: wff set class

Syntax hints: ∧ wa 103 ↔ wb 104 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 ℝcr 7643 0cc0 7644 1c1 7645 · cmul 7649 ℝ^*cxr 7823 < clt 7824 ≤ cle 7825 − cmin 7957 # cap 8367 / cdiv 8456 ℕcn 8744 2c2 8795 3c3 8796 4c4 8797 6c6 8799 ℝ⁺crp 9470 (,]cioc 9702 ↑cexp 10323 √csqrt 10800 sincsin 11387 cosccos 11388 πcpi 11390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 ax-pre-suploc 7765 ax-addf 7766 ax-mulf 7767

This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-disj 3915 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-of 5990 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-oadd 6325 df-er 6437 df-map 6552 df-pm 6553 df-en 6643 df-dom 6644 df-fin 6645 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-xneg 9589 df-xadd 9590 df-ioo 9705 df-ioc 9706 df-ico 9707 df-icc 9708 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-fac 10504 df-bc 10526 df-ihash 10554 df-shft 10619 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 df-sumdc 11155 df-ef 11391 df-sin 11393 df-cos 11394 df-pi 11396 df-rest 12161 df-topgen 12180 df-psmet 12195 df-xmet 12196 df-met 12197 df-bl 12198 df-mopn 12199 df-top 12204 df-topon 12217 df-bases 12249 df-ntr 12304 df-cn 12396 df-cnp 12397 df-tx 12461 df-cncf 12766 df-limced 12833 df-dvap 12834

This theorem is referenced by: pige3 12974

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