cos9thpiminplylem3 - Mathbox for Thierry Arnoux

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Theorem cos9thpiminplylem3 33797

Description: Lemma for cos9thpiminply 33801. (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 11065	. . . . . . . . . 10 ⊢ i ∈ ℂ
3	2	a1i 11	. . . . . . . . 9 ⊢ (⊤ → i ∈ ℂ)
4		2cnd 12203	. . . . . . . . . 10 ⊢ (⊤ → 2 ∈ ℂ)
5		picn 26394	. . . . . . . . . . 11 ⊢ π ∈ ℂ
6	5	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → π ∈ ℂ)
7	4, 6	mulcld 11132	. . . . . . . . 9 ⊢ (⊤ → (2 · π) ∈ ℂ)
8	3, 7	mulcld 11132	. . . . . . . 8 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
9		3cn 12206	. . . . . . . . 9 ⊢ 3 ∈ ℂ
10	9	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℂ)
11		3ne0 12231	. . . . . . . . 9 ⊢ 3 ≠ 0
12	11	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ≠ 0)
13	8, 10, 12	divcld 11897	. . . . . . 7 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
14	13	efcld 15990	. . . . . 6 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
15	1, 14	eqeltrid 2835	. . . . 5 ⊢ (⊤ → 𝑂 ∈ ℂ)
16	15	sqcld 14051	. . . 4 ⊢ (⊤ → (𝑂↑2) ∈ ℂ)
17		1cnd 11107	. . . . 5 ⊢ (⊤ → 1 ∈ ℂ)
18	15, 17	addcld 11131	. . . 4 ⊢ (⊤ → (𝑂 + 1) ∈ ℂ)
19	16, 18	addcomd 11315	. . 3 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = ((𝑂 + 1) + (𝑂↑2)))
20	15, 17	addcomd 11315	. . . . 5 ⊢ (⊤ → (𝑂 + 1) = (1 + 𝑂))
21	20	oveq1d 7361	. . . 4 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = ((1 + 𝑂) + (𝑂↑2)))
22		oveq2 7354	. . . . . 6 ⊢ (𝑛 = 0 → (𝑂↑𝑛) = (𝑂↑0))
23	15	mptru 1548	. . . . . . . 8 ⊢ 𝑂 ∈ ℂ
24	23	a1i 11	. . . . . . 7 ⊢ (𝑛 = 0 → 𝑂 ∈ ℂ)
25	24	exp0d 14047	. . . . . 6 ⊢ (𝑛 = 0 → (𝑂↑0) = 1)
26	22, 25	eqtrd 2766	. . . . 5 ⊢ (𝑛 = 0 → (𝑂↑𝑛) = 1)
27		oveq2 7354	. . . . . 6 ⊢ (𝑛 = 1 → (𝑂↑𝑛) = (𝑂↑1))
28	23	a1i 11	. . . . . . 7 ⊢ (𝑛 = 1 → 𝑂 ∈ ℂ)
29	28	exp1d 14048	. . . . . 6 ⊢ (𝑛 = 1 → (𝑂↑1) = 𝑂)
30	27, 29	eqtrd 2766	. . . . 5 ⊢ (𝑛 = 1 → (𝑂↑𝑛) = 𝑂)
31		oveq2 7354	. . . . 5 ⊢ (𝑛 = 2 → (𝑂↑𝑛) = (𝑂↑2))
32	17, 15, 16	3jca 1128	. . . . 5 ⊢ (⊤ → (1 ∈ ℂ ∧ 𝑂 ∈ ℂ ∧ (𝑂↑2) ∈ ℂ))
33		0cnd 11105	. . . . . 6 ⊢ (⊤ → 0 ∈ ℂ)
34	33, 17, 4	3jca 1128	. . . . 5 ⊢ (⊤ → (0 ∈ ℂ ∧ 1 ∈ ℂ ∧ 2 ∈ ℂ))
35		ax-1ne0 11075	. . . . . . 7 ⊢ 1 ≠ 0
36	35	a1i 11	. . . . . 6 ⊢ (⊤ → 1 ≠ 0)
37	36	necomd 2983	. . . . 5 ⊢ (⊤ → 0 ≠ 1)
38		2ne0 12229	. . . . . . 7 ⊢ 2 ≠ 0
39	38	a1i 11	. . . . . 6 ⊢ (⊤ → 2 ≠ 0)
40	39	necomd 2983	. . . . 5 ⊢ (⊤ → 0 ≠ 2)
41		1ne2 12328	. . . . . 6 ⊢ 1 ≠ 2
42	41	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 2)
43	26, 30, 31, 32, 34, 37, 40, 42	sumtp 15656	. . . 4 ⊢ (⊤ → Σ𝑛 ∈ {0, 1, 2} (𝑂↑𝑛) = ((1 + 𝑂) + (𝑂↑2)))
44		3m1e2 12248	. . . . . . . 8 ⊢ (3 − 1) = 2
45	44	oveq2i 7357	. . . . . . 7 ⊢ (0...(3 − 1)) = (0...2)
46		fz0tp 13528	. . . . . . 7 ⊢ (0...2) = {0, 1, 2}
47	45, 46	eqtri 2754	. . . . . 6 ⊢ (0...(3 − 1)) = {0, 1, 2}
48	47	sumeq1i 15604	. . . . 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 11552	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
51	50	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → i ≠ 0)
52		pine0 26396	. . . . . . . . . . . . . . 15 ⊢ π ≠ 0
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → π ≠ 0)
54	4, 6, 39, 53	mulne0d 11769	. . . . . . . . . . . . 13 ⊢ (⊤ → (2 · π) ≠ 0)
55	3, 7, 51, 54	mulne0d 11769	. . . . . . . . . . . 12 ⊢ (⊤ → (i · (2 · π)) ≠ 0)
56	8, 10, 8, 12, 55	divdiv32d 11922	. . . . . . . . . . 11 ⊢ (⊤ → (((i · (2 · π)) / 3) / (i · (2 · π))) = (((i · (2 · π)) / (i · (2 · π))) / 3))
57	8, 55	dividd 11895	. . . . . . . . . . . 12 ⊢ (⊤ → ((i · (2 · π)) / (i · (2 · π))) = 1)
58	57	oveq1d 7361	. . . . . . . . . . 11 ⊢ (⊤ → (((i · (2 · π)) / (i · (2 · π))) / 3) = (1 / 3))
59	56, 58	eqtrd 2766	. . . . . . . . . 10 ⊢ (⊤ → (((i · (2 · π)) / 3) / (i · (2 · π))) = (1 / 3))
60		3re 12205	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
61	60	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℝ)
62		1lt3 12293	. . . . . . . . . . . 12 ⊢ 1 < 3
63	62	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 1 < 3)
64		recnz 12548	. . . . . . . . . . 11 ⊢ ((3 ∈ ℝ ∧ 1 < 3) → ¬ (1 / 3) ∈ ℤ)
65	61, 63, 64	syl2anc 584	. . . . . . . . . 10 ⊢ (⊤ → ¬ (1 / 3) ∈ ℤ)
66	59, 65	eqneltrd 2851	. . . . . . . . 9 ⊢ (⊤ → ¬ (((i · (2 · π)) / 3) / (i · (2 · π))) ∈ ℤ)
67		efeq1 26464	. . . . . . . . . . 11 ⊢ (((i · (2 · π)) / 3) ∈ ℂ → ((exp‘((i · (2 · π)) / 3)) = 1 ↔ (((i · (2 · π)) / 3) / (i · (2 · π))) ∈ ℤ))
68	67	necon3abid 2964	. . . . . . . . . 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 584	. . . . . . . 8 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 1)
71	49, 70	eqnetrd 2995	. . . . . . 7 ⊢ (⊤ → 𝑂 ≠ 1)
72		3nn0 12399	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
73	72	a1i 11	. . . . . . 7 ⊢ (⊤ → 3 ∈ ℕ₀)
74	15, 71, 73	geoser 15774	. . . . . 6 ⊢ (⊤ → Σ𝑛 ∈ (0...(3 − 1))(𝑂↑𝑛) = ((1 − (𝑂↑3)) / (1 − 𝑂)))
75	49	oveq1d 7361	. . . . . . . . . 10 ⊢ (⊤ → (𝑂↑3) = ((exp‘((i · (2 · π)) / 3))↑3))
76	73	nn0zd 12494	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℤ)
77		efexp 16010	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ (⊤ → (exp‘(3 · ((i · (2 · π)) / 3))) = ((exp‘((i · (2 · π)) / 3))↑3))
79	8, 10, 12	divcan2d 11899	. . . . . . . . . . . 12 ⊢ (⊤ → (3 · ((i · (2 · π)) / 3)) = (i · (2 · π)))
80	79	fveq2d 6826	. . . . . . . . . . 11 ⊢ (⊤ → (exp‘(3 · ((i · (2 · π)) / 3))) = (exp‘(i · (2 · π))))
81		ef2pi 26413	. . . . . . . . . . 11 ⊢ (exp‘(i · (2 · π))) = 1
82	80, 81	eqtrdi 2782	. . . . . . . . . 10 ⊢ (⊤ → (exp‘(3 · ((i · (2 · π)) / 3))) = 1)
83	75, 78, 82	3eqtr2d 2772	. . . . . . . . 9 ⊢ (⊤ → (𝑂↑3) = 1)
84	83	oveq2d 7362	. . . . . . . 8 ⊢ (⊤ → (1 − (𝑂↑3)) = (1 − 1))
85		1m1e0 12197	. . . . . . . 8 ⊢ (1 − 1) = 0
86	84, 85	eqtrdi 2782	. . . . . . 7 ⊢ (⊤ → (1 − (𝑂↑3)) = 0)
87	86	oveq1d 7361	. . . . . 6 ⊢ (⊤ → ((1 − (𝑂↑3)) / (1 − 𝑂)) = (0 / (1 − 𝑂)))
88	17, 15	subcld 11472	. . . . . . 7 ⊢ (⊤ → (1 − 𝑂) ∈ ℂ)
89	71	necomd 2983	. . . . . . . 8 ⊢ (⊤ → 1 ≠ 𝑂)
90	17, 15, 89	subne0d 11481	. . . . . . 7 ⊢ (⊤ → (1 − 𝑂) ≠ 0)
91	88, 90	div0d 11896	. . . . . 6 ⊢ (⊤ → (0 / (1 − 𝑂)) = 0)
92	74, 87, 91	3eqtrd 2770	. . . . 5 ⊢ (⊤ → Σ𝑛 ∈ (0...(3 − 1))(𝑂↑𝑛) = 0)
93	48, 92	eqtr3id 2780	. . . 4 ⊢ (⊤ → Σ𝑛 ∈ {0, 1, 2} (𝑂↑𝑛) = 0)
94	21, 43, 93	3eqtr2d 2772	. . 3 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = 0)
95	19, 94	eqtrd 2766	. 2 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = 0)
96	95	mptru 1548	1 ⊢ ((𝑂↑2) + (𝑂 + 1)) = 0

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1541 ⊤wtru 1542 ∈ wcel 2111 ≠ wne 2928 {ctp 4577 class class class wbr 5089 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 ici 11008 + caddc 11009 · cmul 11011 < clt 11146 − cmin 11344 / cdiv 11774 2c2 12180 3c3 12181 ℕ₀cn0 12381 ℤcz 12468 ...cfz 13407 ↑cexp 13968 Σcsu 15593 expce 15968 πcpi 15973

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-limc 25794 df-dv 25795

This theorem is referenced by: cos9thpiminplylem4 33798 cos9thpinconstrlem1 33802

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