cos9thpinconstrlem1 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cos9thpinconstrlem1		Structured version Visualization version GIF version

Theorem cos9thpinconstrlem1 34284

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 12628	. . . 4 ⊢ (⊤ → 0 ∈ ℤ)
2	1	zconstr 34259	. . 3 ⊢ (⊤ → 0 ∈ Constr)
3		1zzd 12650	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
4	3	zconstr 34259	. . 3 ⊢ (⊤ → 1 ∈ Constr)
5	4	constrnegcl 34258	. . 3 ⊢ (⊤ → -1 ∈ Constr)
6		cos9thpinconstr.1	. . . 4 ⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))
7		ax-icn 11184	. . . . . . . 8 ⊢ i ∈ ℂ
8	7	a1i 11	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
9		2cnd 12344	. . . . . . . 8 ⊢ (⊤ → 2 ∈ ℂ)
10		picn 26689	. . . . . . . . 9 ⊢ π ∈ ℂ
11	10	a1i 11	. . . . . . . 8 ⊢ (⊤ → π ∈ ℂ)
12	9, 11	mulcld 11254	. . . . . . 7 ⊢ (⊤ → (2 · π) ∈ ℂ)
13	8, 12	mulcld 11254	. . . . . 6 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
14		3cn 12347	. . . . . . 7 ⊢ 3 ∈ ℂ
15	14	a1i 11	. . . . . 6 ⊢ (⊤ → 3 ∈ ℂ)
16		3ne0 12375	. . . . . . 7 ⊢ 3 ≠ 0
17	16	a1i 11	. . . . . 6 ⊢ (⊤ → 3 ≠ 0)
18	13, 15, 17	divcld 12016	. . . . 5 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
19	18	efcld 16171	. . . 4 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
20	6, 19	eqeltrid 2866	. . 3 ⊢ (⊤ → 𝑂 ∈ ℂ)
21		0cnd 11224	. . . 4 ⊢ (⊤ → 0 ∈ ℂ)
22	5	constrcn 34255	. . . 4 ⊢ (⊤ → -1 ∈ ℂ)
23		1cnd 11227	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℂ)
24	21, 23	subnegd 11601	. . . . . 6 ⊢ (⊤ → (0 − -1) = (0 + 1))
25	23	addlidd 11436	. . . . . 6 ⊢ (⊤ → (0 + 1) = 1)
26	24, 25	eqtrd 2797	. . . . 5 ⊢ (⊤ → (0 − -1) = 1)
27		ax-1ne0 11194	. . . . . 6 ⊢ 1 ≠ 0
28	27	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 0)
29	26, 28	eqnetrd 3024	. . . 4 ⊢ (⊤ → (0 − -1) ≠ 0)
30	21, 22, 29	subne0ad 11605	. . 3 ⊢ (⊤ → 0 ≠ -1)
31	8, 12, 15, 17	divassd 12051	. . . . . . . 8 ⊢ (⊤ → ((i · (2 · π)) / 3) = (i · ((2 · π) / 3)))
32	31	fveq2d 6886	. . . . . . 7 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) = (exp‘(i · ((2 · π) / 3))))
33	32	fveq2d 6886	. . . . . 6 ⊢ (⊤ → (abs‘(exp‘((i · (2 · π)) / 3))) = (abs‘(exp‘(i · ((2 · π) / 3)))))
34		2re 12340	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
35	34	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 2 ∈ ℝ)
36		pire 26687	. . . . . . . . . 10 ⊢ π ∈ ℝ
37	36	a1i 11	. . . . . . . . 9 ⊢ (⊤ → π ∈ ℝ)
38	35, 37	remulcld 11264	. . . . . . . 8 ⊢ (⊤ → (2 · π) ∈ ℝ)
39		3re 12346	. . . . . . . . 9 ⊢ 3 ∈ ℝ
40	39	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℝ)
41	38, 40, 17	redivcld 12068	. . . . . . 7 ⊢ (⊤ → ((2 · π) / 3) ∈ ℝ)
42		absefi 16286	. . . . . . 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 2797	. . . . 5 ⊢ (⊤ → (abs‘(exp‘((i · (2 · π)) / 3))) = 1)
45	6	a1i 11	. . . . . 6 ⊢ (⊤ → 𝑂 = (exp‘((i · (2 · π)) / 3)))
46	45	fveq2d 6886	. . . . 5 ⊢ (⊤ → (abs‘𝑂) = (abs‘(exp‘((i · (2 · π)) / 3))))
47		1red 11234	. . . . . 6 ⊢ (⊤ → 1 ∈ ℝ)
48		0le1 11762	. . . . . . 7 ⊢ 0 ≤ 1
49	48	a1i 11	. . . . . 6 ⊢ (⊤ → 0 ≤ 1)
50	47, 49	absidd 15510	. . . . 5 ⊢ (⊤ → (abs‘1) = 1)
51	44, 46, 50	3eqtr4d 2807	. . . 4 ⊢ (⊤ → (abs‘𝑂) = (abs‘1))
52	20	subid1d 11583	. . . . 5 ⊢ (⊤ → (𝑂 − 0) = 𝑂)
53	52	fveq2d 6886	. . . 4 ⊢ (⊤ → (abs‘(𝑂 − 0)) = (abs‘𝑂))
54	23	subid1d 11583	. . . . 5 ⊢ (⊤ → (1 − 0) = 1)
55	54	fveq2d 6886	. . . 4 ⊢ (⊤ → (abs‘(1 − 0)) = (abs‘1))
56	51, 53, 55	3eqtr4d 2807	. . 3 ⊢ (⊤ → (abs‘(𝑂 − 0)) = (abs‘(1 − 0)))
57	20, 23	subnegd 11601	. . . . . 6 ⊢ (⊤ → (𝑂 − -1) = (𝑂 + 1))
58	20, 23	addcld 11253	. . . . . . 7 ⊢ (⊤ → (𝑂 + 1) ∈ ℂ)
59	20	sqcld 14208	. . . . . . 7 ⊢ (⊤ → (𝑂↑2) ∈ ℂ)
60	58, 59	addcomd 11437	. . . . . . . 8 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = ((𝑂↑2) + (𝑂 + 1)))
61	6	cos9thpiminplylem3 34279	. . . . . . . . 9 ⊢ ((𝑂↑2) + (𝑂 + 1)) = 0
62	61	a1i 11	. . . . . . . 8 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = 0)
63	60, 62	eqtrd 2797	. . . . . . 7 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = 0)
64		addeq0 11662	. . . . . . . 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 2797	. . . . 5 ⊢ (⊤ → (𝑂 − -1) = -(𝑂↑2))
68	67	fveq2d 6886	. . . 4 ⊢ (⊤ → (abs‘(𝑂 − -1)) = (abs‘-(𝑂↑2)))
69	59	absnegd 15539	. . . 4 ⊢ (⊤ → (abs‘-(𝑂↑2)) = (abs‘(𝑂↑2)))
70		2nn0 12546	. . . . . . 7 ⊢ 2 ∈ ℕ₀
71	70	a1i 11	. . . . . 6 ⊢ (⊤ → 2 ∈ ℕ₀)
72	20, 71	absexpd 15542	. . . . 5 ⊢ (⊤ → (abs‘(𝑂↑2)) = ((abs‘𝑂)↑2))
73	46, 44	eqtrd 2797	. . . . . 6 ⊢ (⊤ → (abs‘𝑂) = 1)
74	73	oveq1d 7431	. . . . 5 ⊢ (⊤ → ((abs‘𝑂)↑2) = (1↑2))
75		sq1 14259	. . . . . 6 ⊢ (1↑2) = 1
76	55, 50	eqtrd 2797	. . . . . 6 ⊢ (⊤ → (abs‘(1 − 0)) = 1)
77	75, 76	eqtr4id 2816	. . . . 5 ⊢ (⊤ → (1↑2) = (abs‘(1 − 0)))
78	72, 74, 77	3eqtrd 2801	. . . 4 ⊢ (⊤ → (abs‘(𝑂↑2)) = (abs‘(1 − 0)))
79	68, 69, 78	3eqtrd 2801	. . 3 ⊢ (⊤ → (abs‘(𝑂 − -1)) = (abs‘(1 − 0)))
80	2, 4, 2, 5, 4, 2, 20, 30, 56, 79	constrcccl 34253	. 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 2957 class class class wbr 5107 ‘cfv 6537 (class class class)co 7416 ℂcc 11123 ℝcr 11124 0cc0 11125 1c1 11126 ici 11127 + caddc 11128 · cmul 11130 ≤ cle 11269 − cmin 11466 -cneg 11467 / cdiv 11896 2c2 12320 3c3 12321 ℕ₀cn0 12529 ↑cexp 14125 abscabs 15321 expce 16149 πcpi 16154 Constrcconstr 34224

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-fac 14338 df-bc 14367 df-hash 14395 df-shft 15140 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-sum 15774 df-ef 16155 df-sin 16157 df-cos 16158 df-pi 16160 df-struct 17241 df-sets 17258 df-slot 17276 df-ndx 17288 df-base 17304 df-ress 17325 df-plusg 17357 df-mulr 17358 df-starv 17359 df-sca 17360 df-vsca 17361 df-ip 17362 df-tset 17363 df-ple 17364 df-ds 17366 df-unif 17367 df-hom 17368 df-cco 17369 df-rest 17509 df-topn 17510 df-0g 17528 df-gsum 17529 df-topgen 17530 df-pt 17531 df-prds 17534 df-xrs 17590 df-qtop 17595 df-imas 17596 df-xps 17598 df-mre 17672 df-mrc 17673 df-acs 17675 df-mgm 18732 df-sgrp 18821 df-mnd 18837 df-submnd 18891 df-mulg 19190 df-cntz 19443 df-cmn 19908 df-psmet 21576 df-xmet 21577 df-met 21578 df-bl 21579 df-mopn 21580 df-fbas 21581 df-fg 21582 df-cnfld 21585 df-top 23118 df-topon 23135 df-topsp 23157 df-bases 23170 df-cld 23243 df-ntr 23244 df-cls 23245 df-nei 23322 df-lp 23360 df-perf 23361 df-cn 23451 df-cnp 23452 df-haus 23539 df-tx 23787 df-hmeo 23980 df-fil 24071 df-fm 24163 df-flim 24164 df-flf 24165 df-xms 24545 df-ms 24546 df-tms 24547 df-cncf 25105 df-limc 26093 df-dv 26094 df-constr 34225

This theorem is used by: cos9thpinconstr 34286

W3C validator