cos9thpiminplylem3 - Mathbox for Thierry Arnoux

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Theorem cos9thpiminplylem3 34205

Description: Lemma for cos9thpiminply 34209. (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 11177	. . . . . . . . . 10 ⊢ i ∈ ℂ
3	2	a1i 11	. . . . . . . . 9 ⊢ (⊤ → i ∈ ℂ)
4		2cnd 12337	. . . . . . . . . 10 ⊢ (⊤ → 2 ∈ ℂ)
5		picn 26658	. . . . . . . . . . 11 ⊢ π ∈ ℂ
6	5	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → π ∈ ℂ)
7	4, 6	mulcld 11247	. . . . . . . . 9 ⊢ (⊤ → (2 · π) ∈ ℂ)
8	3, 7	mulcld 11247	. . . . . . . 8 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
9		3cn 12340	. . . . . . . . 9 ⊢ 3 ∈ ℂ
10	9	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℂ)
11		3ne0 12368	. . . . . . . . 9 ⊢ 3 ≠ 0
12	11	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ≠ 0)
13	8, 10, 12	divcld 12009	. . . . . . 7 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
14	13	efcld 16162	. . . . . 6 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
15	1, 14	eqeltrid 2870	. . . . 5 ⊢ (⊤ → 𝑂 ∈ ℂ)
16	15	sqcld 14200	. . . 4 ⊢ (⊤ → (𝑂↑2) ∈ ℂ)
17		1cnd 11220	. . . . 5 ⊢ (⊤ → 1 ∈ ℂ)
18	15, 17	addcld 11246	. . . 4 ⊢ (⊤ → (𝑂 + 1) ∈ ℂ)
19	16, 18	addcomd 11430	. . 3 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = ((𝑂 + 1) + (𝑂↑2)))
20	15, 17	addcomd 11430	. . . . 5 ⊢ (⊤ → (𝑂 + 1) = (1 + 𝑂))
21	20	oveq1d 7438	. . . 4 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = ((1 + 𝑂) + (𝑂↑2)))
22		oveq2 7431	. . . . . 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 14196	. . . . . 6 ⊢ (𝑛 = 0 → (𝑂↑0) = 1)
26	22, 25	eqtrd 2801	. . . . 5 ⊢ (𝑛 = 0 → (𝑂↑𝑛) = 1)
27		oveq2 7431	. . . . . 6 ⊢ (𝑛 = 1 → (𝑂↑𝑛) = (𝑂↑1))
28	23	a1i 11	. . . . . . 7 ⊢ (𝑛 = 1 → 𝑂 ∈ ℂ)
29	28	exp1d 14197	. . . . . 6 ⊢ (𝑛 = 1 → (𝑂↑1) = 𝑂)
30	27, 29	eqtrd 2801	. . . . 5 ⊢ (𝑛 = 1 → (𝑂↑𝑛) = 𝑂)
31		oveq2 7431	. . . . 5 ⊢ (𝑛 = 2 → (𝑂↑𝑛) = (𝑂↑2))
32	17, 15, 16	3jca 1146	. . . . 5 ⊢ (⊤ → (1 ∈ ℂ ∧ 𝑂 ∈ ℂ ∧ (𝑂↑2) ∈ ℂ))
33		0cnd 11217	. . . . . 6 ⊢ (⊤ → 0 ∈ ℂ)
34	33, 17, 4	3jca 1146	. . . . 5 ⊢ (⊤ → (0 ∈ ℂ ∧ 1 ∈ ℂ ∧ 2 ∈ ℂ))
35		ax-1ne0 11187	. . . . . . 7 ⊢ 1 ≠ 0
36	35	a1i 11	. . . . . 6 ⊢ (⊤ → 1 ≠ 0)
37	36	necomd 3016	. . . . 5 ⊢ (⊤ → 0 ≠ 1)
38		2ne0 12365	. . . . . . 7 ⊢ 2 ≠ 0
39	38	a1i 11	. . . . . 6 ⊢ (⊤ → 2 ≠ 0)
40	39	necomd 3016	. . . . 5 ⊢ (⊤ → 0 ≠ 2)
41		1ne2 12469	. . . . . 6 ⊢ 1 ≠ 2
42	41	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 2)
43	26, 30, 31, 32, 34, 37, 40, 42	sumtp 15826	. . . 4 ⊢ (⊤ → Σ𝑛 ∈ {0, 1, 2} (𝑂↑𝑛) = ((1 + 𝑂) + (𝑂↑2)))
44		3m1e2 12386	. . . . . . . 8 ⊢ (3 − 1) = 2
45	44	oveq2i 7434	. . . . . . 7 ⊢ (0...(3 − 1)) = (0...2)
46		fz0tp 13675	. . . . . . 7 ⊢ (0...2) = {0, 1, 2}
47	45, 46	eqtri 2789	. . . . . 6 ⊢ (0...(3 − 1)) = {0, 1, 2}
48	47	sumeq1i 15774	. . . . 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 11667	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
51	50	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → i ≠ 0)
52		pine0 26662	. . . . . . . . . . . . . . 15 ⊢ π ≠ 0
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → π ≠ 0)
54	4, 6, 39, 53	mulne0d 11884	. . . . . . . . . . . . 13 ⊢ (⊤ → (2 · π) ≠ 0)
55	3, 7, 51, 54	mulne0d 11884	. . . . . . . . . . . 12 ⊢ (⊤ → (i · (2 · π)) ≠ 0)
56	8, 10, 8, 12, 55	divdiv32d 12034	. . . . . . . . . . 11 ⊢ (⊤ → (((i · (2 · π)) / 3) / (i · (2 · π))) = (((i · (2 · π)) / (i · (2 · π))) / 3))
57	8, 55	dividd 12007	. . . . . . . . . . . 12 ⊢ (⊤ → ((i · (2 · π)) / (i · (2 · π))) = 1)
58	57	oveq1d 7438	. . . . . . . . . . 11 ⊢ (⊤ → (((i · (2 · π)) / (i · (2 · π))) / 3) = (1 / 3))
59	56, 58	eqtrd 2801	. . . . . . . . . 10 ⊢ (⊤ → (((i · (2 · π)) / 3) / (i · (2 · π))) = (1 / 3))
60		3re 12339	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
61	60	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℝ)
62		1lt3 12434	. . . . . . . . . . . 12 ⊢ 1 < 3
63	62	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 1 < 3)
64		recnz 12689	. . . . . . . . . . 11 ⊢ ((3 ∈ ℝ ∧ 1 < 3) → ¬ (1 / 3) ∈ ℤ)
65	61, 63, 64	syl2anc 596	. . . . . . . . . 10 ⊢ (⊤ → ¬ (1 / 3) ∈ ℤ)
66	59, 65	eqneltrd 2886	. . . . . . . . 9 ⊢ (⊤ → ¬ (((i · (2 · π)) / 3) / (i · (2 · π))) ∈ ℤ)
67		efeq1 26730	. . . . . . . . . . 11 ⊢ (((i · (2 · π)) / 3) ∈ ℂ → ((exp‘((i · (2 · π)) / 3)) = 1 ↔ (((i · (2 · π)) / 3) / (i · (2 · π))) ∈ ℤ))
68	67	necon3abid 2997	. . . . . . . . . 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 3028	. . . . . . 7 ⊢ (⊤ → 𝑂 ≠ 1)
72		3nn0 12540	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
73	72	a1i 11	. . . . . . 7 ⊢ (⊤ → 3 ∈ ℕ₀)
74	15, 71, 73	geoser 15947	. . . . . 6 ⊢ (⊤ → Σ𝑛 ∈ (0...(3 − 1))(𝑂↑𝑛) = ((1 − (𝑂↑3)) / (1 − 𝑂)))
75	49	oveq1d 7438	. . . . . . . . . 10 ⊢ (⊤ → (𝑂↑3) = ((exp‘((i · (2 · π)) / 3))↑3))
76	73	nn0zd 12634	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℤ)
77		efexp 16182	. . . . . . . . . . 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 12011	. . . . . . . . . . . 12 ⊢ (⊤ → (3 · ((i · (2 · π)) / 3)) = (i · (2 · π)))
80	79	fveq2d 6892	. . . . . . . . . . 11 ⊢ (⊤ → (exp‘(3 · ((i · (2 · π)) / 3))) = (exp‘(i · (2 · π))))
81		ef2pi 26679	. . . . . . . . . . 11 ⊢ (exp‘(i · (2 · π))) = 1
82	80, 81	eqtrdi 2817	. . . . . . . . . 10 ⊢ (⊤ → (exp‘(3 · ((i · (2 · π)) / 3))) = 1)
83	75, 78, 82	3eqtr2d 2807	. . . . . . . . 9 ⊢ (⊤ → (𝑂↑3) = 1)
84	83	oveq2d 7439	. . . . . . . 8 ⊢ (⊤ → (1 − (𝑂↑3)) = (1 − 1))
85		1m1e0 12331	. . . . . . . 8 ⊢ (1 − 1) = 0
86	84, 85	eqtrdi 2817	. . . . . . 7 ⊢ (⊤ → (1 − (𝑂↑3)) = 0)
87	86	oveq1d 7438	. . . . . 6 ⊢ (⊤ → ((1 − (𝑂↑3)) / (1 − 𝑂)) = (0 / (1 − 𝑂)))
88	17, 15	subcld 11587	. . . . . . 7 ⊢ (⊤ → (1 − 𝑂) ∈ ℂ)
89	71	necomd 3016	. . . . . . . 8 ⊢ (⊤ → 1 ≠ 𝑂)
90	17, 15, 89	subne0d 11596	. . . . . . 7 ⊢ (⊤ → (1 − 𝑂) ≠ 0)
91	88, 90	div0d 12008	. . . . . 6 ⊢ (⊤ → (0 / (1 − 𝑂)) = 0)
92	74, 87, 91	3eqtrd 2805	. . . . 5 ⊢ (⊤ → Σ𝑛 ∈ (0...(3 − 1))(𝑂↑𝑛) = 0)
93	48, 92	eqtr3id 2815	. . . 4 ⊢ (⊤ → Σ𝑛 ∈ {0, 1, 2} (𝑂↑𝑛) = 0)
94	21, 43, 93	3eqtr2d 2807	. . 3 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = 0)
95	19, 94	eqtrd 2801	. 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 2146 ≠ wne 2961 {ctp 4598 class class class wbr 5114 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 ℝcr 11117 0cc0 11118 1c1 11119 ici 11120 + caddc 11121 · cmul 11123 < clt 11261 − cmin 11459 / cdiv 11889 2c2 12313 3c3 12314 ℕ₀cn0 12522 ℤcz 12609 ...cfz 13553 ↑cexp 14117 Σcsu 15763 expce 16140 πcpi 16145

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-ioc 13395 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-shft 15130 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-limsup 15548 df-clim 15565 df-rlim 15566 df-sum 15764 df-ef 16146 df-sin 16148 df-cos 16149 df-pi 16151 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-rest 17500 df-topn 17501 df-0g 17519 df-gsum 17520 df-topgen 17521 df-pt 17522 df-prds 17525 df-xrs 17581 df-qtop 17586 df-imas 17587 df-xps 17589 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-submnd 18873 df-mulg 19165 df-cntz 19418 df-cmn 19883 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-fbas 21556 df-fg 21557 df-cnfld 21560 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-cld 23213 df-ntr 23214 df-cls 23215 df-nei 23292 df-lp 23330 df-perf 23331 df-cn 23421 df-cnp 23422 df-haus 23509 df-tx 23756 df-hmeo 23949 df-fil 24040 df-fm 24132 df-flim 24133 df-flf 24134 df-xms 24514 df-ms 24515 df-tms 24516 df-cncf 25074 df-limc 26062 df-dv 26063

This theorem is used by: cos9thpiminplylem4 34206 cos9thpinconstrlem1 34210

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