cos02pilt1 - Metamath Proof Explorer

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Theorem cos02pilt1 26460

Description: Cosine is less than one between zero and 2 · π. (Contributed by Jim Kingdon, 23-Mar-2024.)

**Assertion**
Ref	Expression
cos02pilt1	⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) < 1)

**Proof of Theorem cos02pilt1**
Step	Hyp	Ref	Expression
1		elioore 13272	. . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ ℝ)
2	1	recoscld 16050	. 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) ∈ ℝ)
3		1red 11110	. 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 1 ∈ ℝ)
4		cosbnd 16087	. . . 4 ⊢ (𝐴 ∈ ℝ → (-1 ≤ (cos‘𝐴) ∧ (cos‘𝐴) ≤ 1))
5	4	simprd 495	. . 3 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ≤ 1)
6	1, 5	syl 17	. 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) ≤ 1)
7		0zd 12477	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 ∈ ℤ)
8		2re 12196	. . . . . . . . 9 ⊢ 2 ∈ ℝ
9		pire 26391	. . . . . . . . 9 ⊢ π ∈ ℝ
10	8, 9	remulcli 11125	. . . . . . . 8 ⊢ (2 · π) ∈ ℝ
11	10	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ)
12		0xr 11156	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
13	10	rexri 11167	. . . . . . . . 9 ⊢ (2 · π) ∈ ℝ^*
14		elioo2 13283	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ (2 · π) ∈ ℝ^*) → (𝐴 ∈ (0(,)(2 · π)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (2 · π))))
15	12, 13, 14	mp2an 692	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (2 · π)))
16	15	simp2bi 1146	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < 𝐴)
17		2rp 12892	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
18		pirp 26395	. . . . . . . . 9 ⊢ π ∈ ℝ⁺
19		rpmulcl 12912	. . . . . . . . 9 ⊢ ((2 ∈ ℝ⁺ ∧ π ∈ ℝ⁺) → (2 · π) ∈ ℝ⁺)
20	17, 18, 19	mp2an 692	. . . . . . . 8 ⊢ (2 · π) ∈ ℝ⁺
21		rpgt0 12900	. . . . . . . 8 ⊢ ((2 · π) ∈ ℝ⁺ → 0 < (2 · π))
22	20, 21	mp1i 13	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < (2 · π))
23	1, 11, 16, 22	divgt0d 12054	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < (𝐴 / (2 · π)))
24	20	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ⁺)
25	15	simp3bi 1147	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 < (2 · π))
26	1, 11, 24, 25	ltdiv1dd 12988	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < ((2 · π) / (2 · π)))
27	11	recnd 11137	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℂ)
28	22	gt0ne0d 11678	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ≠ 0)
29	27, 28	dividd 11892	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ((2 · π) / (2 · π)) = 1)
30	26, 29	breqtrd 5117	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < 1)
31		0p1e1 12239	. . . . . . 7 ⊢ (0 + 1) = 1
32	30, 31	breqtrrdi 5133	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < (0 + 1))
33		btwnnz 12546	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 0 < (𝐴 / (2 · π)) ∧ (𝐴 / (2 · π)) < (0 + 1)) → ¬ (𝐴 / (2 · π)) ∈ ℤ)
34	7, 23, 32, 33	syl3anc 1373	. . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ¬ (𝐴 / (2 · π)) ∈ ℤ)
35	1	recnd 11137	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ ℂ)
36		coseq1 26459	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) = 1 ↔ (𝐴 / (2 · π)) ∈ ℤ))
37	35, 36	syl 17	. . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ((cos‘𝐴) = 1 ↔ (𝐴 / (2 · π)) ∈ ℤ))
38	34, 37	mtbird 325	. . . 4 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ¬ (cos‘𝐴) = 1)
39	38	neqned 2935	. . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) ≠ 1)
40	39	necomd 2983	. 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 1 ≠ (cos‘𝐴))
41	2, 3, 6, 40	leneltd 11264	1 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) < 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 class class class wbr 5091 ‘cfv 6481 (class class class)co 7346 ℂcc 11001 ℝcr 11002 0cc0 11003 1c1 11004 + caddc 11006 · cmul 11008 ℝ^*cxr 11142 < clt 11143 ≤ cle 11144 -cneg 11342 / cdiv 11771 2c2 12177 ℤcz 12465 ℝ⁺crp 12887 (,)cioo 13242 cosccos 15968 πcpi 15970

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081 ax-addf 11082

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-q 12844 df-rp 12888 df-xneg 13008 df-xadd 13009 df-xmul 13010 df-ioo 13246 df-ioc 13247 df-ico 13248 df-icc 13249 df-fz 13405 df-fzo 13552 df-fl 13693 df-mod 13771 df-seq 13906 df-exp 13966 df-fac 14178 df-bc 14207 df-hash 14235 df-shft 14971 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-limsup 15375 df-clim 15392 df-rlim 15393 df-sum 15591 df-ef 15971 df-sin 15973 df-cos 15974 df-pi 15976 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-hom 17182 df-cco 17183 df-rest 17323 df-topn 17324 df-0g 17342 df-gsum 17343 df-topgen 17344 df-pt 17345 df-prds 17348 df-xrs 17403 df-qtop 17408 df-imas 17409 df-xps 17411 df-mre 17485 df-mrc 17486 df-acs 17488 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-submnd 18689 df-mulg 18978 df-cntz 19227 df-cmn 19692 df-psmet 21281 df-xmet 21282 df-met 21283 df-bl 21284 df-mopn 21285 df-fbas 21286 df-fg 21287 df-cnfld 21290 df-top 22807 df-topon 22824 df-topsp 22846 df-bases 22859 df-cld 22932 df-ntr 22933 df-cls 22934 df-nei 23011 df-lp 23049 df-perf 23050 df-cn 23140 df-cnp 23141 df-haus 23228 df-tx 23475 df-hmeo 23668 df-fil 23759 df-fm 23851 df-flim 23852 df-flf 23853 df-xms 24233 df-ms 24234 df-tms 24235 df-cncf 24796 df-limc 25792 df-dv 25793

This theorem is referenced by: cosq34lt1 26461 cos0pilt1 26466

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