cos02pilt1 - Metamath Proof Explorer

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Theorem cos02pilt1 26503

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 13303	. . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ ℝ)
2	1	recoscld 16081	. 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) ∈ ℝ)
3		1red 11145	. 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 1 ∈ ℝ)
4		cosbnd 16118	. . . 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 12512	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 ∈ ℤ)
8		2re 12231	. . . . . . . . 9 ⊢ 2 ∈ ℝ
9		pire 26434	. . . . . . . . 9 ⊢ π ∈ ℝ
10	8, 9	remulcli 11160	. . . . . . . 8 ⊢ (2 · π) ∈ ℝ
11	10	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ)
12		0xr 11191	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
13	10	rexri 11202	. . . . . . . . 9 ⊢ (2 · π) ∈ ℝ^*
14		elioo2 13314	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ (2 · π) ∈ ℝ^*) → (𝐴 ∈ (0(,)(2 · π)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (2 · π))))
15	12, 13, 14	mp2an 693	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (2 · π)))
16	15	simp2bi 1147	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < 𝐴)
17		2rp 12922	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
18		pirp 26438	. . . . . . . . 9 ⊢ π ∈ ℝ⁺
19		rpmulcl 12942	. . . . . . . . 9 ⊢ ((2 ∈ ℝ⁺ ∧ π ∈ ℝ⁺) → (2 · π) ∈ ℝ⁺)
20	17, 18, 19	mp2an 693	. . . . . . . 8 ⊢ (2 · π) ∈ ℝ⁺
21		rpgt0 12930	. . . . . . . 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 12089	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 < (𝐴 / (2 · π)))
24	20	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ⁺)
25	15	simp3bi 1148	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 < (2 · π))
26	1, 11, 24, 25	ltdiv1dd 13018	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < ((2 · π) / (2 · π)))
27	11	recnd 11172	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℂ)
28	22	gt0ne0d 11713	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ≠ 0)
29	27, 28	dividd 11927	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ((2 · π) / (2 · π)) = 1)
30	26, 29	breqtrd 5126	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < 1)
31		0p1e1 12274	. . . . . . 7 ⊢ (0 + 1) = 1
32	30, 31	breqtrrdi 5142	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < (0 + 1))
33		btwnnz 12580	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 0 < (𝐴 / (2 · π)) ∧ (𝐴 / (2 · π)) < (0 + 1)) → ¬ (𝐴 / (2 · π)) ∈ ℤ)
34	7, 23, 32, 33	syl3anc 1374	. . . . 5 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ¬ (𝐴 / (2 · π)) ∈ ℤ)
35	1	recnd 11172	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ ℂ)
36		coseq1 26502	. . . . . 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 2940	. . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) ≠ 1)
40	39	necomd 2988	. 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 1 ≠ (cos‘𝐴))
41	2, 3, 6, 40	leneltd 11299	1 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) < 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 -cneg 11377 / cdiv 11806 2c2 12212 ℤcz 12500 ℝ⁺crp 12917 (,)cioo 13273 cosccos 15999 πcpi 16001

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-sum 15622 df-ef 16002 df-sin 16004 df-cos 16005 df-pi 16007 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836

This theorem is referenced by: cosq34lt1 26504 cos0pilt1 26509

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