cos9thpinconstrlem1 - Mathbox for Thierry Arnoux

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Theorem cos9thpinconstrlem1 33933

Description: The complex number 𝑂, representing an angle of (2 · π) / 3, is constructible. (Contributed by Thierry Arnoux, 14-Nov-2025.)

**Hypothesis**
Ref	Expression
cos9thpinconstr.1	⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))

**Assertion**
Ref	Expression
cos9thpinconstrlem1	⊢ 𝑂 ∈ Constr

**Proof of Theorem cos9thpinconstrlem1**
Step	Hyp	Ref	Expression
1		0zd 12536	. . . 4 ⊢ (⊤ → 0 ∈ ℤ)
2	1	zconstr 33908	. . 3 ⊢ (⊤ → 0 ∈ Constr)
3		1zzd 12558	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
4	3	zconstr 33908	. . 3 ⊢ (⊤ → 1 ∈ Constr)
5	4	constrnegcl 33907	. . 3 ⊢ (⊤ → -1 ∈ Constr)
6		cos9thpinconstr.1	. . . 4 ⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))
7		ax-icn 11097	. . . . . . . 8 ⊢ i ∈ ℂ
8	7	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
9		2cnd 12259	. . . . . . . 8 ⊢ (⊤ → 2 ∈ ℂ)
10		picn 26422	. . . . . . . . 9 ⊢ π ∈ ℂ
11	10	a1i 11	. . . . . . . 8 ⊢ (⊤ → π ∈ ℂ)
12	9, 11	mulcld 11165	. . . . . . 7 ⊢ (⊤ → (2 · π) ∈ ℂ)
13	8, 12	mulcld 11165	. . . . . 6 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
14		3cn 12262	. . . . . . 7 ⊢ 3 ∈ ℂ
15	14	a1i 11	. . . . . 6 ⊢ (⊤ → 3 ∈ ℂ)
16		3ne0 12287	. . . . . . 7 ⊢ 3 ≠ 0
17	16	a1i 11	. . . . . 6 ⊢ (⊤ → 3 ≠ 0)
18	13, 15, 17	divcld 11931	. . . . 5 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
19	18	efcld 16048	. . . 4 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
20	6, 19	eqeltrid 2840	. . 3 ⊢ (⊤ → 𝑂 ∈ ℂ)
21		0cnd 11137	. . . 4 ⊢ (⊤ → 0 ∈ ℂ)
22	5	constrcn 33904	. . . 4 ⊢ (⊤ → -1 ∈ ℂ)
23		1cnd 11139	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℂ)
24	21, 23	subnegd 11512	. . . . . 6 ⊢ (⊤ → (0 − -1) = (0 + 1))
25	23	addlidd 11347	. . . . . 6 ⊢ (⊤ → (0 + 1) = 1)
26	24, 25	eqtrd 2771	. . . . 5 ⊢ (⊤ → (0 − -1) = 1)
27		ax-1ne0 11107	. . . . . 6 ⊢ 1 ≠ 0
28	27	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 0)
29	26, 28	eqnetrd 2999	. . . 4 ⊢ (⊤ → (0 − -1) ≠ 0)
30	21, 22, 29	subne0ad 11516	. . 3 ⊢ (⊤ → 0 ≠ -1)
31	8, 12, 15, 17	divassd 11966	. . . . . . . 8 ⊢ (⊤ → ((i · (2 · π)) / 3) = (i · ((2 · π) / 3)))
32	31	fveq2d 6844	. . . . . . 7 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) = (exp‘(i · ((2 · π) / 3))))
33	32	fveq2d 6844	. . . . . 6 ⊢ (⊤ → (abs‘(exp‘((i · (2 · π)) / 3))) = (abs‘(exp‘(i · ((2 · π) / 3)))))
34		2re 12255	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
35	34	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 2 ∈ ℝ)
36		pire 26421	. . . . . . . . . 10 ⊢ π ∈ ℝ
37	36	a1i 11	. . . . . . . . 9 ⊢ (⊤ → π ∈ ℝ)
38	35, 37	remulcld 11175	. . . . . . . 8 ⊢ (⊤ → (2 · π) ∈ ℝ)
39		3re 12261	. . . . . . . . 9 ⊢ 3 ∈ ℝ
40	39	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℝ)
41	38, 40, 17	redivcld 11983	. . . . . . 7 ⊢ (⊤ → ((2 · π) / 3) ∈ ℝ)
42		absefi 16163	. . . . . . 7 ⊢ (((2 · π) / 3) ∈ ℝ → (abs‘(exp‘(i · ((2 · π) / 3)))) = 1)
43	41, 42	syl 17	. . . . . 6 ⊢ (⊤ → (abs‘(exp‘(i · ((2 · π) / 3)))) = 1)
44	33, 43	eqtrd 2771	. . . . 5 ⊢ (⊤ → (abs‘(exp‘((i · (2 · π)) / 3))) = 1)
45	6	a1i 11	. . . . . 6 ⊢ (⊤ → 𝑂 = (exp‘((i · (2 · π)) / 3)))
46	45	fveq2d 6844	. . . . 5 ⊢ (⊤ → (abs‘𝑂) = (abs‘(exp‘((i · (2 · π)) / 3))))
47		1red 11145	. . . . . 6 ⊢ (⊤ → 1 ∈ ℝ)
48		0le1 11673	. . . . . . 7 ⊢ 0 ≤ 1
49	48	a1i 11	. . . . . 6 ⊢ (⊤ → 0 ≤ 1)
50	47, 49	absidd 15385	. . . . 5 ⊢ (⊤ → (abs‘1) = 1)
51	44, 46, 50	3eqtr4d 2781	. . . 4 ⊢ (⊤ → (abs‘𝑂) = (abs‘1))
52	20	subid1d 11494	. . . . 5 ⊢ (⊤ → (𝑂 − 0) = 𝑂)
53	52	fveq2d 6844	. . . 4 ⊢ (⊤ → (abs‘(𝑂 − 0)) = (abs‘𝑂))
54	23	subid1d 11494	. . . . 5 ⊢ (⊤ → (1 − 0) = 1)
55	54	fveq2d 6844	. . . 4 ⊢ (⊤ → (abs‘(1 − 0)) = (abs‘1))
56	51, 53, 55	3eqtr4d 2781	. . 3 ⊢ (⊤ → (abs‘(𝑂 − 0)) = (abs‘(1 − 0)))
57	20, 23	subnegd 11512	. . . . . 6 ⊢ (⊤ → (𝑂 − -1) = (𝑂 + 1))
58	20, 23	addcld 11164	. . . . . . 7 ⊢ (⊤ → (𝑂 + 1) ∈ ℂ)
59	20	sqcld 14106	. . . . . . 7 ⊢ (⊤ → (𝑂↑2) ∈ ℂ)
60	58, 59	addcomd 11348	. . . . . . . 8 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = ((𝑂↑2) + (𝑂 + 1)))
61	6	cos9thpiminplylem3 33928	. . . . . . . . 9 ⊢ ((𝑂↑2) + (𝑂 + 1)) = 0
62	61	a1i 11	. . . . . . . 8 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = 0)
63	60, 62	eqtrd 2771	. . . . . . 7 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = 0)
64		addeq0 11573	. . . . . . . 8 ⊢ (((𝑂 + 1) ∈ ℂ ∧ (𝑂↑2) ∈ ℂ) → (((𝑂 + 1) + (𝑂↑2)) = 0 ↔ (𝑂 + 1) = -(𝑂↑2)))
65	64	biimpa 476	. . . . . . 7 ⊢ ((((𝑂 + 1) ∈ ℂ ∧ (𝑂↑2) ∈ ℂ) ∧ ((𝑂 + 1) + (𝑂↑2)) = 0) → (𝑂 + 1) = -(𝑂↑2))
66	58, 59, 63, 65	syl21anc 838	. . . . . 6 ⊢ (⊤ → (𝑂 + 1) = -(𝑂↑2))
67	57, 66	eqtrd 2771	. . . . 5 ⊢ (⊤ → (𝑂 − -1) = -(𝑂↑2))
68	67	fveq2d 6844	. . . 4 ⊢ (⊤ → (abs‘(𝑂 − -1)) = (abs‘-(𝑂↑2)))
69	59	absnegd 15414	. . . 4 ⊢ (⊤ → (abs‘-(𝑂↑2)) = (abs‘(𝑂↑2)))
70		2nn0 12454	. . . . . . 7 ⊢ 2 ∈ ℕ₀
71	70	a1i 11	. . . . . 6 ⊢ (⊤ → 2 ∈ ℕ₀)
72	20, 71	absexpd 15417	. . . . 5 ⊢ (⊤ → (abs‘(𝑂↑2)) = ((abs‘𝑂)↑2))
73	46, 44	eqtrd 2771	. . . . . 6 ⊢ (⊤ → (abs‘𝑂) = 1)
74	73	oveq1d 7382	. . . . 5 ⊢ (⊤ → ((abs‘𝑂)↑2) = (1↑2))
75		sq1 14157	. . . . . 6 ⊢ (1↑2) = 1
76	55, 50	eqtrd 2771	. . . . . 6 ⊢ (⊤ → (abs‘(1 − 0)) = 1)
77	75, 76	eqtr4id 2790	. . . . 5 ⊢ (⊤ → (1↑2) = (abs‘(1 − 0)))
78	72, 74, 77	3eqtrd 2775	. . . 4 ⊢ (⊤ → (abs‘(𝑂↑2)) = (abs‘(1 − 0)))
79	68, 69, 78	3eqtrd 2775	. . 3 ⊢ (⊤ → (abs‘(𝑂 − -1)) = (abs‘(1 − 0)))
80	2, 4, 2, 5, 4, 2, 20, 30, 56, 79	constrcccl 33902	. 2 ⊢ (⊤ → 𝑂 ∈ Constr)
81	80	mptru 1549	1 ⊢ 𝑂 ∈ Constr

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2932 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 ici 11040 + caddc 11041 · cmul 11043 ≤ cle 11180 − cmin 11377 -cneg 11378 / cdiv 11807 2c2 12236 3c3 12237 ℕ₀cn0 12437 ↑cexp 14023 abscabs 15196 expce 16026 πcpi 16031 Constrcconstr 33873

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-constr 33874

This theorem is referenced by: cos9thpinconstr 33935

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