cos9thpinconstrlem1 - Mathbox for Thierry Arnoux

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Theorem cos9thpinconstrlem1 33895

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 12498	. . . 4 ⊢ (⊤ → 0 ∈ ℤ)
2	1	zconstr 33870	. . 3 ⊢ (⊤ → 0 ∈ Constr)
3		1zzd 12520	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
4	3	zconstr 33870	. . 3 ⊢ (⊤ → 1 ∈ Constr)
5	4	constrnegcl 33869	. . 3 ⊢ (⊤ → -1 ∈ Constr)
6		cos9thpinconstr.1	. . . 4 ⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))
7		ax-icn 11083	. . . . . . . 8 ⊢ i ∈ ℂ
8	7	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
9		2cnd 12221	. . . . . . . 8 ⊢ (⊤ → 2 ∈ ℂ)
10		picn 26421	. . . . . . . . 9 ⊢ π ∈ ℂ
11	10	a1i 11	. . . . . . . 8 ⊢ (⊤ → π ∈ ℂ)
12	9, 11	mulcld 11150	. . . . . . 7 ⊢ (⊤ → (2 · π) ∈ ℂ)
13	8, 12	mulcld 11150	. . . . . 6 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
14		3cn 12224	. . . . . . 7 ⊢ 3 ∈ ℂ
15	14	a1i 11	. . . . . 6 ⊢ (⊤ → 3 ∈ ℂ)
16		3ne0 12249	. . . . . . 7 ⊢ 3 ≠ 0
17	16	a1i 11	. . . . . 6 ⊢ (⊤ → 3 ≠ 0)
18	13, 15, 17	divcld 11915	. . . . 5 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
19	18	efcld 16004	. . . 4 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
20	6, 19	eqeltrid 2838	. . 3 ⊢ (⊤ → 𝑂 ∈ ℂ)
21		0cnd 11123	. . . 4 ⊢ (⊤ → 0 ∈ ℂ)
22	5	constrcn 33866	. . . 4 ⊢ (⊤ → -1 ∈ ℂ)
23		1cnd 11125	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℂ)
24	21, 23	subnegd 11497	. . . . . 6 ⊢ (⊤ → (0 − -1) = (0 + 1))
25	23	addlidd 11332	. . . . . 6 ⊢ (⊤ → (0 + 1) = 1)
26	24, 25	eqtrd 2769	. . . . 5 ⊢ (⊤ → (0 − -1) = 1)
27		ax-1ne0 11093	. . . . . 6 ⊢ 1 ≠ 0
28	27	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 0)
29	26, 28	eqnetrd 2997	. . . 4 ⊢ (⊤ → (0 − -1) ≠ 0)
30	21, 22, 29	subne0ad 11501	. . 3 ⊢ (⊤ → 0 ≠ -1)
31	8, 12, 15, 17	divassd 11950	. . . . . . . 8 ⊢ (⊤ → ((i · (2 · π)) / 3) = (i · ((2 · π) / 3)))
32	31	fveq2d 6836	. . . . . . 7 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) = (exp‘(i · ((2 · π) / 3))))
33	32	fveq2d 6836	. . . . . 6 ⊢ (⊤ → (abs‘(exp‘((i · (2 · π)) / 3))) = (abs‘(exp‘(i · ((2 · π) / 3)))))
34		2re 12217	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
35	34	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 2 ∈ ℝ)
36		pire 26420	. . . . . . . . . 10 ⊢ π ∈ ℝ
37	36	a1i 11	. . . . . . . . 9 ⊢ (⊤ → π ∈ ℝ)
38	35, 37	remulcld 11160	. . . . . . . 8 ⊢ (⊤ → (2 · π) ∈ ℝ)
39		3re 12223	. . . . . . . . 9 ⊢ 3 ∈ ℝ
40	39	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℝ)
41	38, 40, 17	redivcld 11967	. . . . . . 7 ⊢ (⊤ → ((2 · π) / 3) ∈ ℝ)
42		absefi 16119	. . . . . . 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 2769	. . . . 5 ⊢ (⊤ → (abs‘(exp‘((i · (2 · π)) / 3))) = 1)
45	6	a1i 11	. . . . . 6 ⊢ (⊤ → 𝑂 = (exp‘((i · (2 · π)) / 3)))
46	45	fveq2d 6836	. . . . 5 ⊢ (⊤ → (abs‘𝑂) = (abs‘(exp‘((i · (2 · π)) / 3))))
47		1red 11131	. . . . . 6 ⊢ (⊤ → 1 ∈ ℝ)
48		0le1 11658	. . . . . . 7 ⊢ 0 ≤ 1
49	48	a1i 11	. . . . . 6 ⊢ (⊤ → 0 ≤ 1)
50	47, 49	absidd 15344	. . . . 5 ⊢ (⊤ → (abs‘1) = 1)
51	44, 46, 50	3eqtr4d 2779	. . . 4 ⊢ (⊤ → (abs‘𝑂) = (abs‘1))
52	20	subid1d 11479	. . . . 5 ⊢ (⊤ → (𝑂 − 0) = 𝑂)
53	52	fveq2d 6836	. . . 4 ⊢ (⊤ → (abs‘(𝑂 − 0)) = (abs‘𝑂))
54	23	subid1d 11479	. . . . 5 ⊢ (⊤ → (1 − 0) = 1)
55	54	fveq2d 6836	. . . 4 ⊢ (⊤ → (abs‘(1 − 0)) = (abs‘1))
56	51, 53, 55	3eqtr4d 2779	. . 3 ⊢ (⊤ → (abs‘(𝑂 − 0)) = (abs‘(1 − 0)))
57	20, 23	subnegd 11497	. . . . . 6 ⊢ (⊤ → (𝑂 − -1) = (𝑂 + 1))
58	20, 23	addcld 11149	. . . . . . 7 ⊢ (⊤ → (𝑂 + 1) ∈ ℂ)
59	20	sqcld 14065	. . . . . . 7 ⊢ (⊤ → (𝑂↑2) ∈ ℂ)
60	58, 59	addcomd 11333	. . . . . . . 8 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = ((𝑂↑2) + (𝑂 + 1)))
61	6	cos9thpiminplylem3 33890	. . . . . . . . 9 ⊢ ((𝑂↑2) + (𝑂 + 1)) = 0
62	61	a1i 11	. . . . . . . 8 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = 0)
63	60, 62	eqtrd 2769	. . . . . . 7 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = 0)
64		addeq0 11558	. . . . . . . 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 837	. . . . . 6 ⊢ (⊤ → (𝑂 + 1) = -(𝑂↑2))
67	57, 66	eqtrd 2769	. . . . 5 ⊢ (⊤ → (𝑂 − -1) = -(𝑂↑2))
68	67	fveq2d 6836	. . . 4 ⊢ (⊤ → (abs‘(𝑂 − -1)) = (abs‘-(𝑂↑2)))
69	59	absnegd 15373	. . . 4 ⊢ (⊤ → (abs‘-(𝑂↑2)) = (abs‘(𝑂↑2)))
70		2nn0 12416	. . . . . . 7 ⊢ 2 ∈ ℕ₀
71	70	a1i 11	. . . . . 6 ⊢ (⊤ → 2 ∈ ℕ₀)
72	20, 71	absexpd 15376	. . . . 5 ⊢ (⊤ → (abs‘(𝑂↑2)) = ((abs‘𝑂)↑2))
73	46, 44	eqtrd 2769	. . . . . 6 ⊢ (⊤ → (abs‘𝑂) = 1)
74	73	oveq1d 7371	. . . . 5 ⊢ (⊤ → ((abs‘𝑂)↑2) = (1↑2))
75		sq1 14116	. . . . . 6 ⊢ (1↑2) = 1
76	55, 50	eqtrd 2769	. . . . . 6 ⊢ (⊤ → (abs‘(1 − 0)) = 1)
77	75, 76	eqtr4id 2788	. . . . 5 ⊢ (⊤ → (1↑2) = (abs‘(1 − 0)))
78	72, 74, 77	3eqtrd 2773	. . . 4 ⊢ (⊤ → (abs‘(𝑂↑2)) = (abs‘(1 − 0)))
79	68, 69, 78	3eqtrd 2773	. . 3 ⊢ (⊤ → (abs‘(𝑂 − -1)) = (abs‘(1 − 0)))
80	2, 4, 2, 5, 4, 2, 20, 30, 56, 79	constrcccl 33864	. 2 ⊢ (⊤ → 𝑂 ∈ Constr)
81	80	mptru 1548	1 ⊢ 𝑂 ∈ Constr

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1541 ⊤wtru 1542 ∈ wcel 2113 ≠ wne 2930 class class class wbr 5096 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 ℝcr 11023 0cc0 11024 1c1 11025 ici 11026 + caddc 11027 · cmul 11029 ≤ cle 11165 − cmin 11362 -cneg 11363 / cdiv 11792 2c2 12198 3c3 12199 ℕ₀cn0 12399 ↑cexp 13982 abscabs 15155 expce 15982 πcpi 15987 Constrcconstr 33835

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-fac 14195 df-bc 14224 df-hash 14252 df-shft 14988 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410 df-sum 15608 df-ef 15988 df-sin 15990 df-cos 15991 df-pi 15993 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-mulg 18996 df-cntz 19244 df-cmn 19709 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-xms 24262 df-ms 24263 df-tms 24264 df-cncf 24825 df-limc 25821 df-dv 25822 df-constr 33836

This theorem is referenced by: cos9thpinconstr 33897

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