cos9thpinconstrlem1 - Mathbox for Thierry Arnoux

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Theorem cos9thpinconstrlem1 34188

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 12607	. . . 4 ⊢ (⊤ → 0 ∈ ℤ)
2	1	zconstr 34163	. . 3 ⊢ (⊤ → 0 ∈ Constr)
3		1zzd 12629	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
4	3	zconstr 34163	. . 3 ⊢ (⊤ → 1 ∈ Constr)
5	4	constrnegcl 34162	. . 3 ⊢ (⊤ → -1 ∈ Constr)
6		cos9thpinconstr.1	. . . 4 ⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))
7		ax-icn 11163	. . . . . . . 8 ⊢ i ∈ ℂ
8	7	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
9		2cnd 12323	. . . . . . . 8 ⊢ (⊤ → 2 ∈ ℂ)
10		picn 26630	. . . . . . . . 9 ⊢ π ∈ ℂ
11	10	a1i 11	. . . . . . . 8 ⊢ (⊤ → π ∈ ℂ)
12	9, 11	mulcld 11233	. . . . . . 7 ⊢ (⊤ → (2 · π) ∈ ℂ)
13	8, 12	mulcld 11233	. . . . . 6 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
14		3cn 12326	. . . . . . 7 ⊢ 3 ∈ ℂ
15	14	a1i 11	. . . . . 6 ⊢ (⊤ → 3 ∈ ℂ)
16		3ne0 12354	. . . . . . 7 ⊢ 3 ≠ 0
17	16	a1i 11	. . . . . 6 ⊢ (⊤ → 3 ≠ 0)
18	13, 15, 17	divcld 11995	. . . . 5 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
19	18	efcld 16141	. . . 4 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
20	6, 19	eqeltrid 2867	. . 3 ⊢ (⊤ → 𝑂 ∈ ℂ)
21		0cnd 11203	. . . 4 ⊢ (⊤ → 0 ∈ ℂ)
22	5	constrcn 34159	. . . 4 ⊢ (⊤ → -1 ∈ ℂ)
23		1cnd 11206	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℂ)
24	21, 23	subnegd 11580	. . . . . 6 ⊢ (⊤ → (0 − -1) = (0 + 1))
25	23	addlidd 11415	. . . . . 6 ⊢ (⊤ → (0 + 1) = 1)
26	24, 25	eqtrd 2798	. . . . 5 ⊢ (⊤ → (0 − -1) = 1)
27		ax-1ne0 11173	. . . . . 6 ⊢ 1 ≠ 0
28	27	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 0)
29	26, 28	eqnetrd 3025	. . . 4 ⊢ (⊤ → (0 − -1) ≠ 0)
30	21, 22, 29	subne0ad 11584	. . 3 ⊢ (⊤ → 0 ≠ -1)
31	8, 12, 15, 17	divassd 12030	. . . . . . . 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 12319	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
35	34	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 2 ∈ ℝ)
36		pire 26628	. . . . . . . . . 10 ⊢ π ∈ ℝ
37	36	a1i 11	. . . . . . . . 9 ⊢ (⊤ → π ∈ ℝ)
38	35, 37	remulcld 11243	. . . . . . . 8 ⊢ (⊤ → (2 · π) ∈ ℝ)
39		3re 12325	. . . . . . . . 9 ⊢ 3 ∈ ℝ
40	39	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℝ)
41	38, 40, 17	redivcld 12047	. . . . . . 7 ⊢ (⊤ → ((2 · π) / 3) ∈ ℝ)
42		absefi 16256	. . . . . . 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 2798	. . . . 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 11213	. . . . . 6 ⊢ (⊤ → 1 ∈ ℝ)
48		0le1 11741	. . . . . . 7 ⊢ 0 ≤ 1
49	48	a1i 11	. . . . . 6 ⊢ (⊤ → 0 ≤ 1)
50	47, 49	absidd 15479	. . . . 5 ⊢ (⊤ → (abs‘1) = 1)
51	44, 46, 50	3eqtr4d 2808	. . . 4 ⊢ (⊤ → (abs‘𝑂) = (abs‘1))
52	20	subid1d 11562	. . . . 5 ⊢ (⊤ → (𝑂 − 0) = 𝑂)
53	52	fveq2d 6885	. . . 4 ⊢ (⊤ → (abs‘(𝑂 − 0)) = (abs‘𝑂))
54	23	subid1d 11562	. . . . 5 ⊢ (⊤ → (1 − 0) = 1)
55	54	fveq2d 6885	. . . 4 ⊢ (⊤ → (abs‘(1 − 0)) = (abs‘1))
56	51, 53, 55	3eqtr4d 2808	. . 3 ⊢ (⊤ → (abs‘(𝑂 − 0)) = (abs‘(1 − 0)))
57	20, 23	subnegd 11580	. . . . . 6 ⊢ (⊤ → (𝑂 − -1) = (𝑂 + 1))
58	20, 23	addcld 11232	. . . . . . 7 ⊢ (⊤ → (𝑂 + 1) ∈ ℂ)
59	20	sqcld 14185	. . . . . . 7 ⊢ (⊤ → (𝑂↑2) ∈ ℂ)
60	58, 59	addcomd 11416	. . . . . . . 8 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = ((𝑂↑2) + (𝑂 + 1)))
61	6	cos9thpiminplylem3 34183	. . . . . . . . 9 ⊢ ((𝑂↑2) + (𝑂 + 1)) = 0
62	61	a1i 11	. . . . . . . 8 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = 0)
63	60, 62	eqtrd 2798	. . . . . . 7 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = 0)
64		addeq0 11641	. . . . . . . 8 ⊢ (((𝑂 + 1) ∈ ℂ ∧ (𝑂↑2) ∈ ℂ) → (((𝑂 + 1) + (𝑂↑2)) = 0 ↔ (𝑂 + 1) = -(𝑂↑2)))
65	64	biimpa 481	. . . . . . 7 ⊢ ((((𝑂 + 1) ∈ ℂ ∧ (𝑂↑2) ∈ ℂ) ∧ ((𝑂 + 1) + (𝑂↑2)) = 0) → (𝑂 + 1) = -(𝑂↑2))
66	58, 59, 63, 65	syl21anc 850	. . . . . 6 ⊢ (⊤ → (𝑂 + 1) = -(𝑂↑2))
67	57, 66	eqtrd 2798	. . . . 5 ⊢ (⊤ → (𝑂 − -1) = -(𝑂↑2))
68	67	fveq2d 6885	. . . 4 ⊢ (⊤ → (abs‘(𝑂 − -1)) = (abs‘-(𝑂↑2)))
69	59	absnegd 15508	. . . 4 ⊢ (⊤ → (abs‘-(𝑂↑2)) = (abs‘(𝑂↑2)))
70		2nn0 12525	. . . . . . 7 ⊢ 2 ∈ ℕ₀
71	70	a1i 11	. . . . . 6 ⊢ (⊤ → 2 ∈ ℕ₀)
72	20, 71	absexpd 15511	. . . . 5 ⊢ (⊤ → (abs‘(𝑂↑2)) = ((abs‘𝑂)↑2))
73	46, 44	eqtrd 2798	. . . . . 6 ⊢ (⊤ → (abs‘𝑂) = 1)
74	73	oveq1d 7425	. . . . 5 ⊢ (⊤ → ((abs‘𝑂)↑2) = (1↑2))
75		sq1 14236	. . . . . 6 ⊢ (1↑2) = 1
76	55, 50	eqtrd 2798	. . . . . 6 ⊢ (⊤ → (abs‘(1 − 0)) = 1)
77	75, 76	eqtr4id 2817	. . . . 5 ⊢ (⊤ → (1↑2) = (abs‘(1 − 0)))
78	72, 74, 77	3eqtrd 2802	. . . 4 ⊢ (⊤ → (abs‘(𝑂↑2)) = (abs‘(1 − 0)))
79	68, 69, 78	3eqtrd 2802	. . 3 ⊢ (⊤ → (abs‘(𝑂 − -1)) = (abs‘(1 − 0)))
80	2, 4, 2, 5, 4, 2, 20, 30, 56, 79	constrcccl 34157	. 2 ⊢ (⊤ → 𝑂 ∈ Constr)
81	80	mptru 1577	1 ⊢ 𝑂 ∈ Constr

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 400 = wceq 1570 ⊤wtru 1571 ∈ wcel 2143 ≠ wne 2958 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 ℂcc 11102 ℝcr 11103 0cc0 11104 1c1 11105 ici 11106 + caddc 11107 · cmul 11109 ≤ cle 11248 − cmin 11445 -cneg 11446 / cdiv 11875 2c2 12299 3c3 12300 ℕ₀cn0 12508 ↑cexp 14102 abscabs 15290 expce 16119 πcpi 16124 Constrcconstr 34128

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182 ax-addf 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-ioo 13380 df-ioc 13381 df-ico 13382 df-icc 13383 df-fz 13540 df-fzo 13688 df-fl 13830 df-mod 13908 df-seq 14043 df-exp 14103 df-fac 14315 df-bc 14344 df-hash 14372 df-shft 15109 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-limsup 15527 df-clim 15544 df-rlim 15545 df-sum 15743 df-ef 16125 df-sin 16127 df-cos 16128 df-pi 16130 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-starv 17329 df-sca 17330 df-vsca 17331 df-ip 17332 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-hom 17338 df-cco 17339 df-rest 17479 df-topn 17480 df-0g 17498 df-gsum 17499 df-topgen 17500 df-pt 17501 df-prds 17504 df-xrs 17560 df-qtop 17565 df-imas 17566 df-xps 17568 df-mre 17642 df-mrc 17643 df-acs 17645 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-submnd 18846 df-mulg 19138 df-cntz 19391 df-cmn 19856 df-psmet 21523 df-xmet 21524 df-met 21525 df-bl 21526 df-mopn 21527 df-fbas 21528 df-fg 21529 df-cnfld 21532 df-top 23060 df-topon 23077 df-topsp 23099 df-bases 23112 df-cld 23185 df-ntr 23186 df-cls 23187 df-nei 23264 df-lp 23302 df-perf 23303 df-cn 23393 df-cnp 23394 df-haus 23481 df-tx 23728 df-hmeo 23921 df-fil 24012 df-fm 24104 df-flim 24105 df-flf 24106 df-xms 24486 df-ms 24487 df-tms 24488 df-cncf 25046 df-limc 26034 df-dv 26035 df-constr 34129

This theorem is used by: cos9thpinconstr 34190

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