cos9thpiminplylem3 - Mathbox for Thierry Arnoux

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Theorem cos9thpiminplylem3 34327

Description: Lemma for cos9thpiminply 34331. (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 11208	. . . . . . . . . 10 ⊢ i ∈ ℂ
3	2	a1i 11	. . . . . . . . 9 ⊢ (⊤ → i ∈ ℂ)
4		2cnd 12368	. . . . . . . . . 10 ⊢ (⊤ → 2 ∈ ℂ)
5		picn 26726	. . . . . . . . . . 11 ⊢ π ∈ ℂ
6	5	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → π ∈ ℂ)
7	4, 6	mulcld 11278	. . . . . . . . 9 ⊢ (⊤ → (2 · π) ∈ ℂ)
8	3, 7	mulcld 11278	. . . . . . . 8 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
9		3cn 12371	. . . . . . . . 9 ⊢ 3 ∈ ℂ
10	9	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℂ)
11		3ne0 12399	. . . . . . . . 9 ⊢ 3 ≠ 0
12	11	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ≠ 0)
13	8, 10, 12	divcld 12040	. . . . . . 7 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
14	13	efcld 16194	. . . . . 6 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
15	1, 14	eqeltrid 2864	. . . . 5 ⊢ (⊤ → 𝑂 ∈ ℂ)
16	15	sqcld 14233	. . . 4 ⊢ (⊤ → (𝑂↑2) ∈ ℂ)
17		1cnd 11251	. . . . 5 ⊢ (⊤ → 1 ∈ ℂ)
18	15, 17	addcld 11277	. . . 4 ⊢ (⊤ → (𝑂 + 1) ∈ ℂ)
19	16, 18	addcomd 11461	. . 3 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = ((𝑂 + 1) + (𝑂↑2)))
20	15, 17	addcomd 11461	. . . . 5 ⊢ (⊤ → (𝑂 + 1) = (1 + 𝑂))
21	20	oveq1d 7431	. . . 4 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = ((1 + 𝑂) + (𝑂↑2)))
22		oveq2 7424	. . . . . 6 ⊢ (𝑛 = 0 → (𝑂↑𝑛) = (𝑂↑0))
23	15	mptru 1577	. . . . . . . 8 ⊢ 𝑂 ∈ ℂ
24	23	a1i 11	. . . . . . 7 ⊢ (𝑛 = 0 → 𝑂 ∈ ℂ)
25	24	exp0d 14229	. . . . . 6 ⊢ (𝑛 = 0 → (𝑂↑0) = 1)
26	22, 25	eqtrd 2795	. . . . 5 ⊢ (𝑛 = 0 → (𝑂↑𝑛) = 1)
27		oveq2 7424	. . . . . 6 ⊢ (𝑛 = 1 → (𝑂↑𝑛) = (𝑂↑1))
28	23	a1i 11	. . . . . . 7 ⊢ (𝑛 = 1 → 𝑂 ∈ ℂ)
29	28	exp1d 14230	. . . . . 6 ⊢ (𝑛 = 1 → (𝑂↑1) = 𝑂)
30	27, 29	eqtrd 2795	. . . . 5 ⊢ (𝑛 = 1 → (𝑂↑𝑛) = 𝑂)
31		oveq2 7424	. . . . 5 ⊢ (𝑛 = 2 → (𝑂↑𝑛) = (𝑂↑2))
32	17, 15, 16	3jca 1146	. . . . 5 ⊢ (⊤ → (1 ∈ ℂ ∧ 𝑂 ∈ ℂ ∧ (𝑂↑2) ∈ ℂ))
33		0cnd 11248	. . . . . 6 ⊢ (⊤ → 0 ∈ ℂ)
34	33, 17, 4	3jca 1146	. . . . 5 ⊢ (⊤ → (0 ∈ ℂ ∧ 1 ∈ ℂ ∧ 2 ∈ ℂ))
35		ax-1ne0 11218	. . . . . . 7 ⊢ 1 ≠ 0
36	35	a1i 11	. . . . . 6 ⊢ (⊤ → 1 ≠ 0)
37	36	necomd 3010	. . . . 5 ⊢ (⊤ → 0 ≠ 1)
38		2ne0 12396	. . . . . . 7 ⊢ 2 ≠ 0
39	38	a1i 11	. . . . . 6 ⊢ (⊤ → 2 ≠ 0)
40	39	necomd 3010	. . . . 5 ⊢ (⊤ → 0 ≠ 2)
41		1ne2 12500	. . . . . 6 ⊢ 1 ≠ 2
42	41	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 2)
43	26, 30, 31, 32, 34, 37, 40, 42	sumtp 15860	. . . 4 ⊢ (⊤ → Σ𝑛 ∈ {0, 1, 2} (𝑂↑𝑛) = ((1 + 𝑂) + (𝑂↑2)))
44		3m1e2 12417	. . . . . . . 8 ⊢ (3 − 1) = 2
45	44	oveq2i 7427	. . . . . . 7 ⊢ (0...(3 − 1)) = (0...2)
46		fz0tp 13708	. . . . . . 7 ⊢ (0...2) = {0, 1, 2}
47	45, 46	eqtri 2783	. . . . . 6 ⊢ (0...(3 − 1)) = {0, 1, 2}
48	47	sumeq1i 15809	. . . . 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 11698	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
51	50	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → i ≠ 0)
52		pine0 26730	. . . . . . . . . . . . . . 15 ⊢ π ≠ 0
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → π ≠ 0)
54	4, 6, 39, 53	mulne0d 11915	. . . . . . . . . . . . 13 ⊢ (⊤ → (2 · π) ≠ 0)
55	3, 7, 51, 54	mulne0d 11915	. . . . . . . . . . . 12 ⊢ (⊤ → (i · (2 · π)) ≠ 0)
56	8, 10, 8, 12, 55	divdiv32d 12065	. . . . . . . . . . 11 ⊢ (⊤ → (((i · (2 · π)) / 3) / (i · (2 · π))) = (((i · (2 · π)) / (i · (2 · π))) / 3))
57	8, 55	dividd 12038	. . . . . . . . . . . 12 ⊢ (⊤ → ((i · (2 · π)) / (i · (2 · π))) = 1)
58	57	oveq1d 7431	. . . . . . . . . . 11 ⊢ (⊤ → (((i · (2 · π)) / (i · (2 · π))) / 3) = (1 / 3))
59	56, 58	eqtrd 2795	. . . . . . . . . 10 ⊢ (⊤ → (((i · (2 · π)) / 3) / (i · (2 · π))) = (1 / 3))
60		3re 12370	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
61	60	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℝ)
62		1lt3 12465	. . . . . . . . . . . 12 ⊢ 1 < 3
63	62	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 1 < 3)
64		recnz 12721	. . . . . . . . . . 11 ⊢ ((3 ∈ ℝ ∧ 1 < 3) → ¬ (1 / 3) ∈ ℤ)
65	61, 63, 64	syl2anc 596	. . . . . . . . . 10 ⊢ (⊤ → ¬ (1 / 3) ∈ ℤ)
66	59, 65	eqneltrd 2880	. . . . . . . . 9 ⊢ (⊤ → ¬ (((i · (2 · π)) / 3) / (i · (2 · π))) ∈ ℤ)
67		efeq1 26797	. . . . . . . . . . 11 ⊢ (((i · (2 · π)) / 3) ∈ ℂ → ((exp‘((i · (2 · π)) / 3)) = 1 ↔ (((i · (2 · π)) / 3) / (i · (2 · π))) ∈ ℤ))
68	67	necon3abid 2991	. . . . . . . . . 10 ⊢ (((i · (2 · π)) / 3) ∈ ℂ → ((exp‘((i · (2 · π)) / 3)) ≠ 1 ↔ ¬ (((i · (2 · π)) / 3) / (i · (2 · π))) ∈ ℤ))
69	68	biimpar 483	. . . . . . . . 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 596	. . . . . . . 8 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 1)
71	49, 70	eqnetrd 3022	. . . . . . 7 ⊢ (⊤ → 𝑂 ≠ 1)
72		3nn0 12571	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
73	72	a1i 11	. . . . . . 7 ⊢ (⊤ → 3 ∈ ℕ₀)
74	15, 71, 73	geoser 15981	. . . . . 6 ⊢ (⊤ → Σ𝑛 ∈ (0...(3 − 1))(𝑂↑𝑛) = ((1 − (𝑂↑3)) / (1 − 𝑂)))
75	49	oveq1d 7431	. . . . . . . . . 10 ⊢ (⊤ → (𝑂↑3) = ((exp‘((i · (2 · π)) / 3))↑3))
76	73	nn0zd 12665	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℤ)
77		efexp 16214	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ (⊤ → (exp‘(3 · ((i · (2 · π)) / 3))) = ((exp‘((i · (2 · π)) / 3))↑3))
79	8, 10, 12	divcan2d 12042	. . . . . . . . . . . 12 ⊢ (⊤ → (3 · ((i · (2 · π)) / 3)) = (i · (2 · π)))
80	79	fveq2d 6885	. . . . . . . . . . 11 ⊢ (⊤ → (exp‘(3 · ((i · (2 · π)) / 3))) = (exp‘(i · (2 · π))))
81		ef2pi 26747	. . . . . . . . . . 11 ⊢ (exp‘(i · (2 · π))) = 1
82	80, 81	eqtrdi 2811	. . . . . . . . . 10 ⊢ (⊤ → (exp‘(3 · ((i · (2 · π)) / 3))) = 1)
83	75, 78, 82	3eqtr2d 2801	. . . . . . . . 9 ⊢ (⊤ → (𝑂↑3) = 1)
84	83	oveq2d 7432	. . . . . . . 8 ⊢ (⊤ → (1 − (𝑂↑3)) = (1 − 1))
85		1m1e0 12362	. . . . . . . 8 ⊢ (1 − 1) = 0
86	84, 85	eqtrdi 2811	. . . . . . 7 ⊢ (⊤ → (1 − (𝑂↑3)) = 0)
87	86	oveq1d 7431	. . . . . 6 ⊢ (⊤ → ((1 − (𝑂↑3)) / (1 − 𝑂)) = (0 / (1 − 𝑂)))
88	17, 15	subcld 11618	. . . . . . 7 ⊢ (⊤ → (1 − 𝑂) ∈ ℂ)
89	71	necomd 3010	. . . . . . . 8 ⊢ (⊤ → 1 ≠ 𝑂)
90	17, 15, 89	subne0d 11628	. . . . . . 7 ⊢ (⊤ → (1 − 𝑂) ≠ 0)
91	88, 90	div0d 12039	. . . . . 6 ⊢ (⊤ → (0 / (1 − 𝑂)) = 0)
92	74, 87, 91	3eqtrd 2799	. . . . 5 ⊢ (⊤ → Σ𝑛 ∈ (0...(3 − 1))(𝑂↑𝑛) = 0)
93	48, 92	eqtr3id 2809	. . . 4 ⊢ (⊤ → Σ𝑛 ∈ {0, 1, 2} (𝑂↑𝑛) = 0)
94	21, 43, 93	3eqtr2d 2801	. . 3 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = 0)
95	19, 94	eqtrd 2795	. 2 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = 0)
96	95	mptru 1577	1 ⊢ ((𝑂↑2) + (𝑂 + 1)) = 0

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 ≠ wne 2955 {ctp 4588 class class class wbr 5103 ‘cfv 6535 (class class class)co 7416 ℂcc 11147 ℝcr 11148 0cc0 11149 1c1 11150 ici 11151 + caddc 11152 · cmul 11154 < clt 11292 − cmin 11490 / cdiv 11920 2c2 12344 3c3 12345 ℕ₀cn0 12553 ℤcz 12640 ...cfz 13586 ↑cexp 14150 Σcsu 15798 expce 16172 πcpi 16177

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 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-inf2 9627 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9339 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9489 df-card 9969 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-q 13023 df-rp 13068 df-xneg 13188 df-xadd 13189 df-xmul 13190 df-ioo 13427 df-ioc 13428 df-ico 13429 df-icc 13430 df-fz 13587 df-fzo 13735 df-fl 13878 df-mod 13956 df-seq 14091 df-exp 14151 df-fac 14363 df-bc 14392 df-hash 14420 df-shft 15165 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-limsup 15583 df-clim 15600 df-rlim 15601 df-sum 15799 df-ef 16178 df-sin 16180 df-cos 16181 df-pi 16183 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-starv 17382 df-sca 17383 df-vsca 17384 df-ip 17385 df-tset 17386 df-ple 17387 df-ds 17389 df-unif 17390 df-hom 17391 df-cco 17392 df-rest 17532 df-topn 17533 df-0g 17551 df-gsum 17552 df-topgen 17553 df-pt 17554 df-prds 17557 df-xrs 17613 df-qtop 17618 df-imas 17619 df-xps 17621 df-mre 17695 df-mrc 17696 df-acs 17698 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-submnd 18918 df-mulg 19217 df-cntz 19470 df-cmn 19935 df-psmet 21609 df-xmet 21610 df-met 21611 df-bl 21612 df-mopn 21613 df-fbas 21614 df-fg 21615 df-cnfld 21618 df-top 23151 df-topon 23168 df-topsp 23190 df-bases 23203 df-cld 23276 df-ntr 23277 df-cls 23278 df-nei 23355 df-lp 23393 df-perf 23394 df-cn 23484 df-cnp 23485 df-haus 23572 df-tx 23820 df-hmeo 24013 df-fil 24104 df-fm 24196 df-flim 24197 df-flf 24198 df-xms 24578 df-ms 24579 df-tms 24580 df-cncf 25138 df-limc 26125 df-dv 26126

This theorem is used by: cos9thpiminplylem4 34328 cos9thpinconstrlem1 34332

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