cos9thpiminplylem3 - Mathbox for Thierry Arnoux

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Theorem cos9thpiminplylem3 33943

Description: Lemma for cos9thpiminply 33947. (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 11089	. . . . . . . . . 10 ⊢ i ∈ ℂ
3	2	a1i 11	. . . . . . . . 9 ⊢ (⊤ → i ∈ ℂ)
4		2cnd 12227	. . . . . . . . . 10 ⊢ (⊤ → 2 ∈ ℂ)
5		picn 26427	. . . . . . . . . . 11 ⊢ π ∈ ℂ
6	5	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → π ∈ ℂ)
7	4, 6	mulcld 11156	. . . . . . . . 9 ⊢ (⊤ → (2 · π) ∈ ℂ)
8	3, 7	mulcld 11156	. . . . . . . 8 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ)
9		3cn 12230	. . . . . . . . 9 ⊢ 3 ∈ ℂ
10	9	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ∈ ℂ)
11		3ne0 12255	. . . . . . . . 9 ⊢ 3 ≠ 0
12	11	a1i 11	. . . . . . . 8 ⊢ (⊤ → 3 ≠ 0)
13	8, 10, 12	divcld 11921	. . . . . . 7 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ)
14	13	efcld 16010	. . . . . 6 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ)
15	1, 14	eqeltrid 2841	. . . . 5 ⊢ (⊤ → 𝑂 ∈ ℂ)
16	15	sqcld 14071	. . . 4 ⊢ (⊤ → (𝑂↑2) ∈ ℂ)
17		1cnd 11131	. . . . 5 ⊢ (⊤ → 1 ∈ ℂ)
18	15, 17	addcld 11155	. . . 4 ⊢ (⊤ → (𝑂 + 1) ∈ ℂ)
19	16, 18	addcomd 11339	. . 3 ⊢ (⊤ → ((𝑂↑2) + (𝑂 + 1)) = ((𝑂 + 1) + (𝑂↑2)))
20	15, 17	addcomd 11339	. . . . 5 ⊢ (⊤ → (𝑂 + 1) = (1 + 𝑂))
21	20	oveq1d 7375	. . . 4 ⊢ (⊤ → ((𝑂 + 1) + (𝑂↑2)) = ((1 + 𝑂) + (𝑂↑2)))
22		oveq2 7368	. . . . . 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 14067	. . . . . 6 ⊢ (𝑛 = 0 → (𝑂↑0) = 1)
26	22, 25	eqtrd 2772	. . . . 5 ⊢ (𝑛 = 0 → (𝑂↑𝑛) = 1)
27		oveq2 7368	. . . . . 6 ⊢ (𝑛 = 1 → (𝑂↑𝑛) = (𝑂↑1))
28	23	a1i 11	. . . . . . 7 ⊢ (𝑛 = 1 → 𝑂 ∈ ℂ)
29	28	exp1d 14068	. . . . . 6 ⊢ (𝑛 = 1 → (𝑂↑1) = 𝑂)
30	27, 29	eqtrd 2772	. . . . 5 ⊢ (𝑛 = 1 → (𝑂↑𝑛) = 𝑂)
31		oveq2 7368	. . . . 5 ⊢ (𝑛 = 2 → (𝑂↑𝑛) = (𝑂↑2))
32	17, 15, 16	3jca 1129	. . . . 5 ⊢ (⊤ → (1 ∈ ℂ ∧ 𝑂 ∈ ℂ ∧ (𝑂↑2) ∈ ℂ))
33		0cnd 11129	. . . . . 6 ⊢ (⊤ → 0 ∈ ℂ)
34	33, 17, 4	3jca 1129	. . . . 5 ⊢ (⊤ → (0 ∈ ℂ ∧ 1 ∈ ℂ ∧ 2 ∈ ℂ))
35		ax-1ne0 11099	. . . . . . 7 ⊢ 1 ≠ 0
36	35	a1i 11	. . . . . 6 ⊢ (⊤ → 1 ≠ 0)
37	36	necomd 2988	. . . . 5 ⊢ (⊤ → 0 ≠ 1)
38		2ne0 12253	. . . . . . 7 ⊢ 2 ≠ 0
39	38	a1i 11	. . . . . 6 ⊢ (⊤ → 2 ≠ 0)
40	39	necomd 2988	. . . . 5 ⊢ (⊤ → 0 ≠ 2)
41		1ne2 12352	. . . . . 6 ⊢ 1 ≠ 2
42	41	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 2)
43	26, 30, 31, 32, 34, 37, 40, 42	sumtp 15676	. . . 4 ⊢ (⊤ → Σ𝑛 ∈ {0, 1, 2} (𝑂↑𝑛) = ((1 + 𝑂) + (𝑂↑2)))
44		3m1e2 12272	. . . . . . . 8 ⊢ (3 − 1) = 2
45	44	oveq2i 7371	. . . . . . 7 ⊢ (0...(3 − 1)) = (0...2)
46		fz0tp 13548	. . . . . . 7 ⊢ (0...2) = {0, 1, 2}
47	45, 46	eqtri 2760	. . . . . 6 ⊢ (0...(3 − 1)) = {0, 1, 2}
48	47	sumeq1i 15624	. . . . 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 11576	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
51	50	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → i ≠ 0)
52		pine0 26429	. . . . . . . . . . . . . . 15 ⊢ π ≠ 0
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (⊤ → π ≠ 0)
54	4, 6, 39, 53	mulne0d 11793	. . . . . . . . . . . . 13 ⊢ (⊤ → (2 · π) ≠ 0)
55	3, 7, 51, 54	mulne0d 11793	. . . . . . . . . . . 12 ⊢ (⊤ → (i · (2 · π)) ≠ 0)
56	8, 10, 8, 12, 55	divdiv32d 11946	. . . . . . . . . . 11 ⊢ (⊤ → (((i · (2 · π)) / 3) / (i · (2 · π))) = (((i · (2 · π)) / (i · (2 · π))) / 3))
57	8, 55	dividd 11919	. . . . . . . . . . . 12 ⊢ (⊤ → ((i · (2 · π)) / (i · (2 · π))) = 1)
58	57	oveq1d 7375	. . . . . . . . . . 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 12229	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
61	60	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℝ)
62		1lt3 12317	. . . . . . . . . . . 12 ⊢ 1 < 3
63	62	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 1 < 3)
64		recnz 12571	. . . . . . . . . . 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 26497	. . . . . . . . . . 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 12423	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
73	72	a1i 11	. . . . . . 7 ⊢ (⊤ → 3 ∈ ℕ₀)
74	15, 71, 73	geoser 15794	. . . . . 6 ⊢ (⊤ → Σ𝑛 ∈ (0...(3 − 1))(𝑂↑𝑛) = ((1 − (𝑂↑3)) / (1 − 𝑂)))
75	49	oveq1d 7375	. . . . . . . . . 10 ⊢ (⊤ → (𝑂↑3) = ((exp‘((i · (2 · π)) / 3))↑3))
76	73	nn0zd 12517	. . . . . . . . . . 11 ⊢ (⊤ → 3 ∈ ℤ)
77		efexp 16030	. . . . . . . . . . 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 11923	. . . . . . . . . . . 12 ⊢ (⊤ → (3 · ((i · (2 · π)) / 3)) = (i · (2 · π)))
80	79	fveq2d 6839	. . . . . . . . . . 11 ⊢ (⊤ → (exp‘(3 · ((i · (2 · π)) / 3))) = (exp‘(i · (2 · π))))
81		ef2pi 26446	. . . . . . . . . . 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 7376	. . . . . . . 8 ⊢ (⊤ → (1 − (𝑂↑3)) = (1 − 1))
85		1m1e0 12221	. . . . . . . 8 ⊢ (1 − 1) = 0
86	84, 85	eqtrdi 2788	. . . . . . 7 ⊢ (⊤ → (1 − (𝑂↑3)) = 0)
87	86	oveq1d 7375	. . . . . 6 ⊢ (⊤ → ((1 − (𝑂↑3)) / (1 − 𝑂)) = (0 / (1 − 𝑂)))
88	17, 15	subcld 11496	. . . . . . 7 ⊢ (⊤ → (1 − 𝑂) ∈ ℂ)
89	71	necomd 2988	. . . . . . . 8 ⊢ (⊤ → 1 ≠ 𝑂)
90	17, 15, 89	subne0d 11505	. . . . . . 7 ⊢ (⊤ → (1 − 𝑂) ≠ 0)
91	88, 90	div0d 11920	. . . . . 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 4585 class class class wbr 5099 ‘cfv 6493 (class class class)co 7360 ℂcc 11028 ℝcr 11029 0cc0 11030 1c1 11031 ici 11032 + caddc 11033 · cmul 11035 < clt 11170 − cmin 11368 / cdiv 11798 2c2 12204 3c3 12205 ℕ₀cn0 12405 ℤcz 12492 ...cfz 13427 ↑cexp 13988 Σcsu 15613 expce 15988 πcpi 15993

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

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 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ioc 13270 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-fac 14201 df-bc 14230 df-hash 14258 df-shft 14994 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-limsup 15398 df-clim 15415 df-rlim 15416 df-sum 15614 df-ef 15994 df-sin 15996 df-cos 15997 df-pi 15999 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24268 df-ms 24269 df-tms 24270 df-cncf 24831 df-limc 25827 df-dv 25828

This theorem is referenced by: cos9thpiminplylem4 33944 cos9thpinconstrlem1 33948

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