cos9thpinconstrlem1 - Mathbox for Thierry Arnoux

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Theorem cos9thpinconstrlem1 34332

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 12652	. . . 4 ⊢ (⊤ → 0 ∈ ℤ)
2	1	zconstr 34307	. . 3 ⊢ (⊤ → 0 ∈ Constr)
3		1zzd 12674	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
4	3	zconstr 34307	. . 3 ⊢ (⊤ → 1 ∈ Constr)
5	4	constrnegcl 34306	. . 3 ⊢ (⊤ → -1 ∈ Constr)
6		cos9thpinconstr.1	. . . 4 ⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))
7		ax-icn 11208	. . . . . . . 8 ⊢ i ∈ ℂ
8	7	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
9		2cnd 12368	. . . . . . . 8 ⊢ (⊤ → 2 ∈ ℂ)
10		picn 26726	. . . . . . . . 9 ⊢ π ∈ ℂ
11	10	a1i 11	. . . . . . . 8 ⊢ (⊤ → π ∈ ℂ)
12	9, 11	mulcld 11278	. . . . . . 7 ⊢ (⊤ → (2 · π) ∈ ℂ)
13	8, 12	mulcld 11278	. . . . . 6 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
14		3cn 12371	. . . . . . 7 ⊢ 3 ∈ ℂ
15	14	a1i 11	. . . . . 6 ⊢ (⊤ → 3 ∈ ℂ)
16		3ne0 12399	. . . . . . 7 ⊢ 3 ≠ 0
17	16	a1i 11	. . . . . 6 ⊢ (⊤ → 3 ≠ 0)
18	13, 15, 17	divcld 12040	. . . . 5 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
19	18	efcld 16194	. . . 4 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
20	6, 19	eqeltrid 2864	. . 3 ⊢ (⊤ → 𝑂 ∈ ℂ)
21		0cnd 11248	. . . 4 ⊢ (⊤ → 0 ∈ ℂ)
22	5	constrcn 34303	. . . 4 ⊢ (⊤ → -1 ∈ ℂ)
23		1cnd 11251	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℂ)
24	21, 23	subnegd 11625	. . . . . 6 ⊢ (⊤ → (0 − -1) = (0 + 1))
25	23	addlidd 11460	. . . . . 6 ⊢ (⊤ → (0 + 1) = 1)
26	24, 25	eqtrd 2795	. . . . 5 ⊢ (⊤ → (0 − -1) = 1)
27		ax-1ne0 11218	. . . . . 6 ⊢ 1 ≠ 0
28	27	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 0)
29	26, 28	eqnetrd 3022	. . . 4 ⊢ (⊤ → (0 − -1) ≠ 0)
30	21, 22, 29	subne0ad 11629	. . 3 ⊢ (⊤ → 0 ≠ -1)
31	8, 12, 15, 17	divassd 12075	. . . . . . . 8 ⊢ (⊤ → ((i · (2 · π)) / 3) = (i · ((2 · π) / 3)))
32	31	fveq2d 6885	. . . . . . 7 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) = (exp‘(i · ((2 · π) / 3))))
33	32	fveq2d 6885	. . . . . 6 ⊢ (⊤ → (abs‘(exp‘((i · (2 · π)) / 3))) = (abs‘(exp‘(i · ((2 · π) / 3)))))
34		2re 12364	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
35	34	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 2 ∈ ℝ)
36		pire 26724	. . . . . . . . . 10 ⊢ π ∈ ℝ
37	36	a1i 11	. . . . . . . . 9 ⊢ (⊤ → π ∈ ℝ)
38	35, 37	remulcld 11288	. . . . . . . 8 ⊢ (⊤ → (2 · π) ∈ ℝ)
39		3re 12370	. . . . . . . . 9 ⊢ 3 ∈ ℝ
40	39	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℝ)
41	38, 40, 17	redivcld 12092	. . . . . . 7 ⊢ (⊤ → ((2 · π) / 3) ∈ ℝ)
42		absefi 16309	. . . . . . 7 ⊢ (((2 · π) / 3) ∈ ℝ → (abs‘(exp‘(i · ((2 · π) / 3)))) = 1)
43	41, 42	syl 18	. . . . . 6 ⊢ (⊤ → (abs‘(exp‘(i · ((2 · π) / 3)))) = 1)
44	33, 43	eqtrd 2795	. . . . 5 ⊢ (⊤ → (abs‘(exp‘((i · (2 · π)) / 3))) = 1)
45	6	a1i 11	. . . . . 6 ⊢ (⊤ → 𝑂 = (exp‘((i · (2 · π)) / 3)))
46	45	fveq2d 6885	. . . . 5 ⊢ (⊤ → (abs‘𝑂) = (abs‘(exp‘((i · (2 · π)) / 3))))
47		1red 11258	. . . . . 6 ⊢ (⊤ → 1 ∈ ℝ)
48		0le1 11786	. . . . . . 7 ⊢ 0 ≤ 1
49	48	a1i 11	. . . . . 6 ⊢ (⊤ → 0 ≤ 1)
50	47, 49	absidd 15535	. . . . 5 ⊢ (⊤ → (abs‘1) = 1)
51	44, 46, 50	3eqtr4d 2805	. . . 4 ⊢ (⊤ → (abs‘𝑂) = (abs‘1))
52	20	subid1d 11607	. . . . 5 ⊢ (⊤ → (𝑂 − 0) = 𝑂)
53	52	fveq2d 6885	. . . 4 ⊢ (⊤ → (abs‘(𝑂 − 0)) = (abs‘𝑂))
54	23	subid1d 11607	. . . . 5 ⊢ (⊤ → (1 − 0) = 1)
55	54	fveq2d 6885	. . . 4 ⊢ (⊤ → (abs‘(1 − 0)) = (abs‘1))
56	51, 53, 55	3eqtr4d 2805	. . 3 ⊢ (⊤ → (abs‘(𝑂 − 0)) = (abs‘(1 − 0)))
57	20, 23	subnegd 11625	. . . . . 6 ⊢ (⊤ → (𝑂 − -1) = (𝑂 + 1))
58	20, 23	addcld 11277	. . . . . . 7 ⊢ (⊤ → (𝑂 + 1) ∈ ℂ)
59	20	sqcld 14233	. . . . . . 7 ⊢ (⊤ → (𝑂↑2) ∈ ℂ)
60	58, 59	addcomd 11461	. . . . . . . 8 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = ((𝑂↑2) + (𝑂 + 1)))
61	6	cos9thpiminplylem3 34327	. . . . . . . . 9 ⊢ ((𝑂↑2) + (𝑂 + 1)) = 0
62	61	a1i 11	. . . . . . . 8 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = 0)
63	60, 62	eqtrd 2795	. . . . . . 7 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = 0)
64		addeq0 11686	. . . . . . . 8 ⊢ (((𝑂 + 1) ∈ ℂ ∧ (𝑂↑2) ∈ ℂ) → (((𝑂 + 1) + (𝑂↑2)) = 0 ↔ (𝑂 + 1) = -(𝑂↑2)))
65	64	biimpa 482	. . . . . . 7 ⊢ ((((𝑂 + 1) ∈ ℂ ∧ (𝑂↑2) ∈ ℂ) ∧ ((𝑂 + 1) + (𝑂↑2)) = 0) → (𝑂 + 1) = -(𝑂↑2))
66	58, 59, 63, 65	syl21anc 851	. . . . . 6 ⊢ (⊤ → (𝑂 + 1) = -(𝑂↑2))
67	57, 66	eqtrd 2795	. . . . 5 ⊢ (⊤ → (𝑂 − -1) = -(𝑂↑2))
68	67	fveq2d 6885	. . . 4 ⊢ (⊤ → (abs‘(𝑂 − -1)) = (abs‘-(𝑂↑2)))
69	59	absnegd 15564	. . . 4 ⊢ (⊤ → (abs‘-(𝑂↑2)) = (abs‘(𝑂↑2)))
70		2nn0 12570	. . . . . . 7 ⊢ 2 ∈ ℕ₀
71	70	a1i 11	. . . . . 6 ⊢ (⊤ → 2 ∈ ℕ₀)
72	20, 71	absexpd 15567	. . . . 5 ⊢ (⊤ → (abs‘(𝑂↑2)) = ((abs‘𝑂)↑2))
73	46, 44	eqtrd 2795	. . . . . 6 ⊢ (⊤ → (abs‘𝑂) = 1)
74	73	oveq1d 7431	. . . . 5 ⊢ (⊤ → ((abs‘𝑂)↑2) = (1↑2))
75		sq1 14284	. . . . . 6 ⊢ (1↑2) = 1
76	55, 50	eqtrd 2795	. . . . . 6 ⊢ (⊤ → (abs‘(1 − 0)) = 1)
77	75, 76	eqtr4id 2814	. . . . 5 ⊢ (⊤ → (1↑2) = (abs‘(1 − 0)))
78	72, 74, 77	3eqtrd 2799	. . . 4 ⊢ (⊤ → (abs‘(𝑂↑2)) = (abs‘(1 − 0)))
79	68, 69, 78	3eqtrd 2799	. . 3 ⊢ (⊤ → (abs‘(𝑂 − -1)) = (abs‘(1 − 0)))
80	2, 4, 2, 5, 4, 2, 20, 30, 56, 79	constrcccl 34301	. 2 ⊢ (⊤ → 𝑂 ∈ Constr)
81	80	mptru 1577	1 ⊢ 𝑂 ∈ Constr

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 ≠ wne 2955 class class class wbr 5103 ‘cfv 6535 (class class class)co 7416 ℂcc 11147 ℝcr 11148 0cc0 11149 1c1 11150 ici 11151 + caddc 11152 · cmul 11154 ≤ cle 11293 − cmin 11490 -cneg 11491 / cdiv 11920 2c2 12344 3c3 12345 ℕ₀cn0 12553 ↑cexp 14150 abscabs 15346 expce 16172 πcpi 16177 Constrcconstr 34272

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-inf2 9627 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9339 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9489 df-card 9969 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-q 13023 df-rp 13068 df-xneg 13188 df-xadd 13189 df-xmul 13190 df-ioo 13427 df-ioc 13428 df-ico 13429 df-icc 13430 df-fz 13587 df-fzo 13735 df-fl 13878 df-mod 13956 df-seq 14091 df-exp 14151 df-fac 14363 df-bc 14392 df-hash 14420 df-shft 15165 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-limsup 15583 df-clim 15600 df-rlim 15601 df-sum 15799 df-ef 16178 df-sin 16180 df-cos 16181 df-pi 16183 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-starv 17382 df-sca 17383 df-vsca 17384 df-ip 17385 df-tset 17386 df-ple 17387 df-ds 17389 df-unif 17390 df-hom 17391 df-cco 17392 df-rest 17532 df-topn 17533 df-0g 17551 df-gsum 17552 df-topgen 17553 df-pt 17554 df-prds 17557 df-xrs 17613 df-qtop 17618 df-imas 17619 df-xps 17621 df-mre 17695 df-mrc 17696 df-acs 17698 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-submnd 18918 df-mulg 19217 df-cntz 19470 df-cmn 19935 df-psmet 21609 df-xmet 21610 df-met 21611 df-bl 21612 df-mopn 21613 df-fbas 21614 df-fg 21615 df-cnfld 21618 df-top 23151 df-topon 23168 df-topsp 23190 df-bases 23203 df-cld 23276 df-ntr 23277 df-cls 23278 df-nei 23355 df-lp 23393 df-perf 23394 df-cn 23484 df-cnp 23485 df-haus 23572 df-tx 23820 df-hmeo 24013 df-fil 24104 df-fm 24196 df-flim 24197 df-flf 24198 df-xms 24578 df-ms 24579 df-tms 24580 df-cncf 25138 df-limc 26125 df-dv 26126 df-constr 34273

This theorem is used by: cos9thpinconstr 34334

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