pigt3 - Intuitionistic Logic Explorer

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Theorem pigt3 16037

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 16035	. . . . 5 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))
2	1	simpli 111	. . . 4 ⊢ (sin‘(π / 6)) = (1 / 2)
3		ax-1cn 8273	. . . . 5 ⊢ 1 ∈ ℂ
4		2cn 9378	. . . . . 6 ⊢ 2 ∈ ℂ
5		2ap0 9400	. . . . . 6 ⊢ 2 # 0
6	4, 5	pm3.2i 272	. . . . 5 ⊢ (2 ∈ ℂ ∧ 2 # 0)
7		3cn 9382	. . . . . 6 ⊢ 3 ∈ ℂ
8		3ap0 9403	. . . . . 6 ⊢ 3 # 0
9	7, 8	pm3.2i 272	. . . . 5 ⊢ (3 ∈ ℂ ∧ 3 # 0)
10		divcanap5 9047	. . . . 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 9463	. . . . 5 ⊢ (3 · 1) = 3
13		3t2e6 9464	. . . . 5 ⊢ (3 · 2) = 6
14	12, 13	oveq12i 6097	. . . 4 ⊢ ((3 · 1) / (3 · 2)) = (3 / 6)
15	2, 11, 14	3eqtr2i 2265	. . 3 ⊢ (sin‘(π / 6)) = (3 / 6)
16		pire 15979	. . . . . . 7 ⊢ π ∈ ℝ
17		6nn 9475	. . . . . . 7 ⊢ 6 ∈ ℕ
18		nndivre 9343	. . . . . . 7 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ)
19	16, 17, 18	mp2an 430	. . . . . 6 ⊢ (π / 6) ∈ ℝ
20		6re 9388	. . . . . . 7 ⊢ 6 ∈ ℝ
21		pipos 15981	. . . . . . 7 ⊢ 0 < π
22		6pos 9408	. . . . . . 7 ⊢ 0 < 6
23	16, 20, 21, 22	divgt0ii 9252	. . . . . 6 ⊢ 0 < (π / 6)
24		1re 8326	. . . . . . 7 ⊢ 1 ∈ ℝ
25		pigt2lt4 15977	. . . . . . . . . 10 ⊢ (2 < π ∧ π < 4)
26	25	simpri 113	. . . . . . . . 9 ⊢ π < 4
27		4re 9384	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
28	16, 27, 20, 22	ltdiv1ii 9262	. . . . . . . . 9 ⊢ (π < 4 ↔ (π / 6) < (4 / 6))
29	26, 28	mpbi 145	. . . . . . . 8 ⊢ (π / 6) < (4 / 6)
30		4lt6 9490	. . . . . . . . 9 ⊢ 4 < 6
31	20, 22	elrpii 10068	. . . . . . . . . 10 ⊢ 6 ∈ ℝ⁺
32		divlt1lt 10136	. . . . . . . . . 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 9343	. . . . . . . . . 10 ⊢ ((4 ∈ ℝ ∧ 6 ∈ ℕ) → (4 / 6) ∈ ℝ)
36	27, 17, 35	mp2an 430	. . . . . . . . 9 ⊢ (4 / 6) ∈ ℝ
37	19, 36, 24	lttri 8432	. . . . . . . 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 8430	. . . . . 6 ⊢ (π / 6) ≤ 1
40		0xr 8373	. . . . . . 7 ⊢ 0 ∈ ℝ^*
41		elioc2 10349	. . . . . . 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 12543	. . . . 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 4153	. 2 ⊢ (3 / 6) < (π / 6)
48		3re 9381	. . 3 ⊢ 3 ∈ ℝ
49	48, 16, 20, 22	ltdiv1ii 9262	. 2 ⊢ (3 < π ↔ (3 / 6) < (π / 6))
50	47, 49	mpbir 146	1 ⊢ 3 < π

Colors of variables: wff set class

This proof depends on syntax axioms: ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 ℂcc 8178 ℝcr 8179 0cc0 8180 1c1 8181 · cmul 8185 ℝ^*cxr 8360 < clt 8361 ≤ cle 8362 − cmin 8499 # cap 8912 / cdiv 9005 ℕcn 9307 2c2 9358 3c3 9359 4c4 9360 6c6 9362 ℝ⁺crp 10065 (,]cioc 10302 ↑cexp 10990 √csqrt 11778 sincsin 12430 cosccos 12431 πcpi 12433

This proof depends on 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 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299 ax-caucvg 8300 ax-pre-suploc 8301 ax-addf 8302 ax-mulf 8303

This proof 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 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-disj 4107 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-of 6302 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-map 6924 df-pm 6925 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7325 df-inf 7326 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-5 9369 df-6 9370 df-7 9371 df-8 9372 df-9 9373 df-n0 9569 df-z 9650 df-uz 9932 df-q 10030 df-rp 10066 df-xneg 10185 df-xadd 10186 df-ioo 10305 df-ioc 10306 df-ico 10307 df-icc 10308 df-fz 10423 df-fzo 10561 df-seqfrec 10900 df-exp 10991 df-fac 11180 df-bc 11202 df-ihash 11231 df-shft 11596 df-cj 11623 df-re 11624 df-im 11625 df-rsqrt 11780 df-abs 11781 df-clim 12064 df-sumdc 12139 df-ef 12434 df-sin 12436 df-cos 12437 df-pi 12439 df-rest 13648 df-topgen 13667 df-psmet 14964 df-xmet 14965 df-met 14966 df-bl 14967 df-mopn 14968 df-top 15190 df-topon 15203 df-bases 15235 df-ntr 15288 df-cn 15380 df-cnp 15381 df-tx 15445 df-cncf 15763 df-limced 15848 df-dvap 15849

This theorem is used by: pige3 16038

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