cos9thpiminplylem3 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cos9thpiminplylem3		Structured version Visualization version GIF version

Theorem cos9thpiminplylem3 33948

Description: Lemma for cos9thpiminply 33952. (Contributed by Thierry Arnoux, 14-Nov-2025.)

**Hypothesis**
Ref	Expression
cos9thpiminplylem3.1	⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))

**Assertion**
Ref	Expression
cos9thpiminplylem3	⊢ ((𝑂↑2) + (𝑂 + 1)) = 0

Proof of Theorem cos9thpiminplylem3 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cos9thpiminplylem3.1	. . . . . 6 ⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))
2		ax-icn 11092	. . . . . . . . . 10 ⊢ i ∈ ℂ
3	2	a1i 11	. . . . . . . . 9 ⊢ (⊤ → i ∈ ℂ)
4		2cnd 12254	. . . . . . . . . 10 ⊢ (⊤ → 2 ∈ ℂ)
5		picn 26439	. . . . . . . . . . 11 ⊢ π ∈ ℂ
6	5	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → π ∈ ℂ)
7	4, 6	mulcld 11160	. . . . . . . . 9 ⊢ (⊤ → (2 · π) ∈ ℂ)
8	3, 7	mulcld 11160	. . . . . . . 8 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
9		3cn 12257	. . . . . . . . 9 ⊢ 3 ∈ ℂ
10	9	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℂ)
11		3ne0 12282	. . . . . . . . 9 ⊢ 3 ≠ 0
12	11	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ≠ 0)
13	8, 10, 12	divcld 11926	. . . . . . 7 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
14	13	efcld 16043	. . . . . 6 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
15	1, 14	eqeltrid 2841	. . . . 5 ⊢ (⊤ → 𝑂 ∈ ℂ)
16	15	sqcld 14101	. . . 4 ⊢ (⊤ → (𝑂↑2) ∈ ℂ)
17		1cnd 11134	. . . . 5 ⊢ (⊤ → 1 ∈ ℂ)
18	15, 17	addcld 11159	. . . 4 ⊢ (⊤ → (𝑂 + 1) ∈ ℂ)
19	16, 18	addcomd 11343	. . 3 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = ((𝑂 + 1) + (𝑂↑2)))
20	15, 17	addcomd 11343	. . . . 5 ⊢ (⊤ → (𝑂 + 1) = (1 + 𝑂))
21	20	oveq1d 7377	. . . 4 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = ((1 + 𝑂) + (𝑂↑2)))
22		oveq2 7370	. . . . . 6 ⊢ (𝑛 = 0 → (𝑂↑𝑛) = (𝑂↑0))
23	15	mptru 1549	. . . . . . . 8 ⊢ 𝑂 ∈ ℂ
24	23	a1i 11	. . . . . . 7 ⊢ (𝑛 = 0 → 𝑂 ∈ ℂ)
25	24	exp0d 14097	. . . . . 6 ⊢ (𝑛 = 0 → (𝑂↑0) = 1)
26	22, 25	eqtrd 2772	. . . . 5 ⊢ (𝑛 = 0 → (𝑂↑𝑛) = 1)
27		oveq2 7370	. . . . . 6 ⊢ (𝑛 = 1 → (𝑂↑𝑛) = (𝑂↑1))
28	23	a1i 11	. . . . . . 7 ⊢ (𝑛 = 1 → 𝑂 ∈ ℂ)
29	28	exp1d 14098	. . . . . 6 ⊢ (𝑛 = 1 → (𝑂↑1) = 𝑂)
30	27, 29	eqtrd 2772	. . . . 5 ⊢ (𝑛 = 1 → (𝑂↑𝑛) = 𝑂)
31		oveq2 7370	. . . . 5 ⊢ (𝑛 = 2 → (𝑂↑𝑛) = (𝑂↑2))
32	17, 15, 16	3jca 1129	. . . . 5 ⊢ (⊤ → (1 ∈ ℂ ∧ 𝑂 ∈ ℂ ∧ (𝑂↑2) ∈ ℂ))
33		0cnd 11132	. . . . . 6 ⊢ (⊤ → 0 ∈ ℂ)
34	33, 17, 4	3jca 1129	. . . . 5 ⊢ (⊤ → (0 ∈ ℂ ∧ 1 ∈ ℂ ∧ 2 ∈ ℂ))
35		ax-1ne0 11102	. . . . . . 7 ⊢ 1 ≠ 0
36	35	a1i 11	. . . . . 6 ⊢ (⊤ → 1 ≠ 0)
37	36	necomd 2988	. . . . 5 ⊢ (⊤ → 0 ≠ 1)
38		2ne0 12280	. . . . . . 7 ⊢ 2 ≠ 0
39	38	a1i 11	. . . . . 6 ⊢ (⊤ → 2 ≠ 0)
40	39	necomd 2988	. . . . 5 ⊢ (⊤ → 0 ≠ 2)
41		1ne2 12379	. . . . . 6 ⊢ 1 ≠ 2
42	41	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 2)
43	26, 30, 31, 32, 34, 37, 40, 42	sumtp 15706	. . . 4 ⊢ (⊤ → Σ𝑛 ∈ {0, 1, 2} (𝑂↑𝑛) = ((1 + 𝑂) + (𝑂↑2)))
44		3m1e2 12299	. . . . . . . 8 ⊢ (3 − 1) = 2
45	44	oveq2i 7373	. . . . . . 7 ⊢ (0...(3 − 1)) = (0...2)
46		fz0tp 13577	. . . . . . 7 ⊢ (0...2) = {0, 1, 2}
47	45, 46	eqtri 2760	. . . . . 6 ⊢ (0...(3 − 1)) = {0, 1, 2}
48	47	sumeq1i 15654	. . . . 5 ⊢ Σ𝑛 ∈ (0...(3 − 1))(𝑂↑𝑛) = Σ𝑛 ∈ {0, 1, 2} (𝑂↑𝑛)
49	1	a1i 11	. . . . . . . 8 ⊢ (⊤ → 𝑂 = (exp‘((i · (2 · π)) / 3)))
50		ine0 11580	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
51	50	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → i ≠ 0)
52		pine0 26441	. . . . . . . . . . . . . . 15 ⊢ π ≠ 0
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → π ≠ 0)
54	4, 6, 39, 53	mulne0d 11797	. . . . . . . . . . . . 13 ⊢ (⊤ → (2 · π) ≠ 0)
55	3, 7, 51, 54	mulne0d 11797	. . . . . . . . . . . 12 ⊢ (⊤ → (i · (2 · π)) ≠ 0)
56	8, 10, 8, 12, 55	divdiv32d 11951	. . . . . . . . . . 11 ⊢ (⊤ → (((i · (2 · π)) / 3) / (i · (2 · π))) = (((i · (2 · π)) / (i · (2 · π))) / 3))
57	8, 55	dividd 11924	. . . . . . . . . . . 12 ⊢ (⊤ → ((i · (2 · π)) / (i · (2 · π))) = 1)
58	57	oveq1d 7377	. . . . . . . . . . 11 ⊢ (⊤ → (((i · (2 · π)) / (i · (2 · π))) / 3) = (1 / 3))
59	56, 58	eqtrd 2772	. . . . . . . . . 10 ⊢ (⊤ → (((i · (2 · π)) / 3) / (i · (2 · π))) = (1 / 3))
60		3re 12256	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
61	60	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℝ)
62		1lt3 12344	. . . . . . . . . . . 12 ⊢ 1 < 3
63	62	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 1 < 3)
64		recnz 12599	. . . . . . . . . . 11 ⊢ ((3 ∈ ℝ ∧ 1 < 3) → ¬ (1 / 3) ∈ ℤ)
65	61, 63, 64	syl2anc 585	. . . . . . . . . 10 ⊢ (⊤ → ¬ (1 / 3) ∈ ℤ)
66	59, 65	eqneltrd 2857	. . . . . . . . 9 ⊢ (⊤ → ¬ (((i · (2 · π)) / 3) / (i · (2 · π))) ∈ ℤ)
67		efeq1 26509	. . . . . . . . . . 11 ⊢ (((i · (2 · π)) / 3) ∈ ℂ → ((exp‘((i · (2 · π)) / 3)) = 1 ↔ (((i · (2 · π)) / 3) / (i · (2 · π))) ∈ ℤ))
68	67	necon3abid 2969	. . . . . . . . . 10 ⊢ (((i · (2 · π)) / 3) ∈ ℂ → ((exp‘((i · (2 · π)) / 3)) ≠ 1 ↔ ¬ (((i · (2 · π)) / 3) / (i · (2 · π))) ∈ ℤ))
69	68	biimpar 477	. . . . . . . . 9 ⊢ ((((i · (2 · π)) / 3) ∈ ℂ ∧ ¬ (((i · (2 · π)) / 3) / (i · (2 · π))) ∈ ℤ) → (exp‘((i · (2 · π)) / 3)) ≠ 1)
70	13, 66, 69	syl2anc 585	. . . . . . . 8 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 1)
71	49, 70	eqnetrd 3000	. . . . . . 7 ⊢ (⊤ → 𝑂 ≠ 1)
72		3nn0 12450	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
73	72	a1i 11	. . . . . . 7 ⊢ (⊤ → 3 ∈ ℕ₀)
74	15, 71, 73	geoser 15827	. . . . . 6 ⊢ (⊤ → Σ𝑛 ∈ (0...(3 − 1))(𝑂↑𝑛) = ((1 − (𝑂↑3)) / (1 − 𝑂)))
75	49	oveq1d 7377	. . . . . . . . . 10 ⊢ (⊤ → (𝑂↑3) = ((exp‘((i · (2 · π)) / 3))↑3))
76	73	nn0zd 12544	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℤ)
77		efexp 16063	. . . . . . . . . . 11 ⊢ ((((i · (2 · π)) / 3) ∈ ℂ ∧ 3 ∈ ℤ) → (exp‘(3 · ((i · (2 · π)) / 3))) = ((exp‘((i · (2 · π)) / 3))↑3))
78	13, 76, 77	syl2anc 585	. . . . . . . . . 10 ⊢ (⊤ → (exp‘(3 · ((i · (2 · π)) / 3))) = ((exp‘((i · (2 · π)) / 3))↑3))
79	8, 10, 12	divcan2d 11928	. . . . . . . . . . . 12 ⊢ (⊤ → (3 · ((i · (2 · π)) / 3)) = (i · (2 · π)))
80	79	fveq2d 6840	. . . . . . . . . . 11 ⊢ (⊤ → (exp‘(3 · ((i · (2 · π)) / 3))) = (exp‘(i · (2 · π))))
81		ef2pi 26458	. . . . . . . . . . 11 ⊢ (exp‘(i · (2 · π))) = 1
82	80, 81	eqtrdi 2788	. . . . . . . . . 10 ⊢ (⊤ → (exp‘(3 · ((i · (2 · π)) / 3))) = 1)
83	75, 78, 82	3eqtr2d 2778	. . . . . . . . 9 ⊢ (⊤ → (𝑂↑3) = 1)
84	83	oveq2d 7378	. . . . . . . 8 ⊢ (⊤ → (1 − (𝑂↑3)) = (1 − 1))
85		1m1e0 12248	. . . . . . . 8 ⊢ (1 − 1) = 0
86	84, 85	eqtrdi 2788	. . . . . . 7 ⊢ (⊤ → (1 − (𝑂↑3)) = 0)
87	86	oveq1d 7377	. . . . . 6 ⊢ (⊤ → ((1 − (𝑂↑3)) / (1 − 𝑂)) = (0 / (1 − 𝑂)))
88	17, 15	subcld 11500	. . . . . . 7 ⊢ (⊤ → (1 − 𝑂) ∈ ℂ)
89	71	necomd 2988	. . . . . . . 8 ⊢ (⊤ → 1 ≠ 𝑂)
90	17, 15, 89	subne0d 11509	. . . . . . 7 ⊢ (⊤ → (1 − 𝑂) ≠ 0)
91	88, 90	div0d 11925	. . . . . 6 ⊢ (⊤ → (0 / (1 − 𝑂)) = 0)
92	74, 87, 91	3eqtrd 2776	. . . . 5 ⊢ (⊤ → Σ𝑛 ∈ (0...(3 − 1))(𝑂↑𝑛) = 0)
93	48, 92	eqtr3id 2786	. . . 4 ⊢ (⊤ → Σ𝑛 ∈ {0, 1, 2} (𝑂↑𝑛) = 0)
94	21, 43, 93	3eqtr2d 2778	. . 3 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = 0)
95	19, 94	eqtrd 2772	. 2 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = 0)
96	95	mptru 1549	1 ⊢ ((𝑂↑2) + (𝑂 + 1)) = 0

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2933 {ctp 4572 class class class wbr 5086 ‘cfv 6494 (class class class)co 7362 ℂcc 11031 ℝcr 11032 0cc0 11033 1c1 11034 ici 11035 + caddc 11036 · cmul 11038 < clt 11174 − cmin 11372 / cdiv 11802 2c2 12231 3c3 12232 ℕ₀cn0 12432 ℤcz 12519 ...cfz 13456 ↑cexp 14018 Σcsu 15643 expce 16021 πcpi 16026

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 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7626 df-om 7813 df-1st 7937 df-2nd 7938 df-supp 8106 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-er 8638 df-map 8770 df-pm 8771 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-fi 9319 df-sup 9350 df-inf 9351 df-oi 9420 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ioc 13298 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15024 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-limsup 15428 df-clim 15445 df-rlim 15446 df-sum 15644 df-ef 16027 df-sin 16029 df-cos 16030 df-pi 16032 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-mulg 19039 df-cntz 19287 df-cmn 19752 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-fbas 21345 df-fg 21346 df-cnfld 21349 df-top 22873 df-topon 22890 df-topsp 22912 df-bases 22925 df-cld 22998 df-ntr 22999 df-cls 23000 df-nei 23077 df-lp 23115 df-perf 23116 df-cn 23206 df-cnp 23207 df-haus 23294 df-tx 23541 df-hmeo 23734 df-fil 23825 df-fm 23917 df-flim 23918 df-flf 23919 df-xms 24299 df-ms 24300 df-tms 24301 df-cncf 24859 df-limc 25847 df-dv 25848

This theorem is referenced by: cos9thpiminplylem4 33949 cos9thpinconstrlem1 33953

W3C validator