cos02pilt1 - Metamath Proof Explorer

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Theorem cos02pilt1 26468

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 13281	. . 3 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ ℝ)
2	1	recoscld 16059	. 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) ∈ ℝ)
3		1red 11119	. 2 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 1 ∈ ℝ)
4		cosbnd 16096	. . . 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 12486	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 0 ∈ ℤ)
8		2re 12205	. . . . . . . . 9 ⊢ 2 ∈ ℝ
9		pire 26399	. . . . . . . . 9 ⊢ π ∈ ℝ
10	8, 9	remulcli 11134	. . . . . . . 8 ⊢ (2 · π) ∈ ℝ
11	10	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℝ)
12		0xr 11165	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
13	10	rexri 11176	. . . . . . . . 9 ⊢ (2 · π) ∈ ℝ^*
14		elioo2 13292	. . . . . . . . 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 12901	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
18		pirp 26403	. . . . . . . . 9 ⊢ π ∈ ℝ⁺
19		rpmulcl 12921	. . . . . . . . 9 ⊢ ((2 ∈ ℝ⁺ ∧ π ∈ ℝ⁺) → (2 · π) ∈ ℝ⁺)
20	17, 18, 19	mp2an 692	. . . . . . . 8 ⊢ (2 · π) ∈ ℝ⁺
21		rpgt0 12909	. . . . . . . 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 12063	. . . . . 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 12997	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < ((2 · π) / (2 · π)))
27	11	recnd 11146	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ∈ ℂ)
28	22	gt0ne0d 11687	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (2 · π) ≠ 0)
29	27, 28	dividd 11901	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(2 · π)) → ((2 · π) / (2 · π)) = 1)
30	26, 29	breqtrd 5119	. . . . . . 7 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < 1)
31		0p1e1 12248	. . . . . . 7 ⊢ (0 + 1) = 1
32	30, 31	breqtrrdi 5135	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → (𝐴 / (2 · π)) < (0 + 1))
33		btwnnz 12555	. . . . . 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 11146	. . . . . 6 ⊢ (𝐴 ∈ (0(,)(2 · π)) → 𝐴 ∈ ℂ)
36		coseq1 26467	. . . . . 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 11273	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 5093 ‘cfv 6487 (class class class)co 7352 ℂcc 11010 ℝcr 11011 0cc0 11012 1c1 11013 + caddc 11015 · cmul 11017 ℝ^*cxr 11151 < clt 11152 ≤ cle 11153 -cneg 11351 / cdiv 11780 2c2 12186 ℤcz 12474 ℝ⁺crp 12896 (,)cioo 13251 cosccos 15977 πcpi 15979

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 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9537 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-pre-sup 11090 ax-addf 11091

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 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9838 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-div 11781 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-z 12475 df-dec 12595 df-uz 12739 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13255 df-ioc 13256 df-ico 13257 df-icc 13258 df-fz 13414 df-fzo 13561 df-fl 13702 df-mod 13780 df-seq 13915 df-exp 13975 df-fac 14187 df-bc 14216 df-hash 14244 df-shft 14980 df-cj 15012 df-re 15013 df-im 15014 df-sqrt 15148 df-abs 15149 df-limsup 15384 df-clim 15401 df-rlim 15402 df-sum 15600 df-ef 15980 df-sin 15982 df-cos 15983 df-pi 15985 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-starv 17182 df-sca 17183 df-vsca 17184 df-ip 17185 df-tset 17186 df-ple 17187 df-ds 17189 df-unif 17190 df-hom 17191 df-cco 17192 df-rest 17332 df-topn 17333 df-0g 17351 df-gsum 17352 df-topgen 17353 df-pt 17354 df-prds 17357 df-xrs 17412 df-qtop 17417 df-imas 17418 df-xps 17420 df-mre 17494 df-mrc 17495 df-acs 17497 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-submnd 18698 df-mulg 18987 df-cntz 19235 df-cmn 19700 df-psmet 21289 df-xmet 21290 df-met 21291 df-bl 21292 df-mopn 21293 df-fbas 21294 df-fg 21295 df-cnfld 21298 df-top 22815 df-topon 22832 df-topsp 22854 df-bases 22867 df-cld 22940 df-ntr 22941 df-cls 22942 df-nei 23019 df-lp 23057 df-perf 23058 df-cn 23148 df-cnp 23149 df-haus 23236 df-tx 23483 df-hmeo 23676 df-fil 23767 df-fm 23859 df-flim 23860 df-flf 23861 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-limc 25800 df-dv 25801

This theorem is referenced by: cosq34lt1 26469 cos0pilt1 26474

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