1cubrlem - Metamath Proof Explorer

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Theorem 1cubrlem 26582

Description: The cube roots of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)

**Assertion**
Ref	Expression
1cubrlem	⊢ ((-1↑_𝑐(2 / 3)) = ((-1 + (i · (√‘3))) / 2) ∧ ((-1↑_𝑐(2 / 3))↑2) = ((-1 − (i · (√‘3))) / 2))

**Proof of Theorem 1cubrlem**
Step	Hyp	Ref	Expression
1		neg1cn 12330	. . . 4 ⊢ -1 ∈ ℂ
2		neg1ne0 12332	. . . 4 ⊢ -1 ≠ 0
3		2re 12290	. . . . . 6 ⊢ 2 ∈ ℝ
4		3nn 12295	. . . . . 6 ⊢ 3 ∈ ℕ
5		nndivre 12257	. . . . . 6 ⊢ ((2 ∈ ℝ ∧ 3 ∈ ℕ) → (2 / 3) ∈ ℝ)
6	3, 4, 5	mp2an 688	. . . . 5 ⊢ (2 / 3) ∈ ℝ
7	6	recni 11232	. . . 4 ⊢ (2 / 3) ∈ ℂ
8		cxpef 26409	. . . 4 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0 ∧ (2 / 3) ∈ ℂ) → (-1↑_𝑐(2 / 3)) = (exp‘((2 / 3) · (log‘-1))))
9	1, 2, 7, 8	mp3an 1459	. . 3 ⊢ (-1↑_𝑐(2 / 3)) = (exp‘((2 / 3) · (log‘-1)))
10		logm1 26333	. . . . . 6 ⊢ (log‘-1) = (i · π)
11	10	oveq2i 7422	. . . . 5 ⊢ ((2 / 3) · (log‘-1)) = ((2 / 3) · (i · π))
12		ax-icn 11171	. . . . . 6 ⊢ i ∈ ℂ
13		pire 26204	. . . . . . 7 ⊢ π ∈ ℝ
14	13	recni 11232	. . . . . 6 ⊢ π ∈ ℂ
15	7, 12, 14	mul12i 11413	. . . . 5 ⊢ ((2 / 3) · (i · π)) = (i · ((2 / 3) · π))
16	11, 15	eqtri 2758	. . . 4 ⊢ ((2 / 3) · (log‘-1)) = (i · ((2 / 3) · π))
17	16	fveq2i 6893	. . 3 ⊢ (exp‘((2 / 3) · (log‘-1))) = (exp‘(i · ((2 / 3) · π)))
18		6nn 12305	. . . . . . . . 9 ⊢ 6 ∈ ℕ
19		nndivre 12257	. . . . . . . . 9 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ)
20	13, 18, 19	mp2an 688	. . . . . . . 8 ⊢ (π / 6) ∈ ℝ
21	20	recni 11232	. . . . . . 7 ⊢ (π / 6) ∈ ℂ
22		coshalfpip 26240	. . . . . . 7 ⊢ ((π / 6) ∈ ℂ → (cos‘((π / 2) + (π / 6))) = -(sin‘(π / 6)))
23	21, 22	ax-mp 5	. . . . . 6 ⊢ (cos‘((π / 2) + (π / 6))) = -(sin‘(π / 6))
24		2cn 12291	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
25		2ne0 12320	. . . . . . . . . 10 ⊢ 2 ≠ 0
26		divrec2 11893	. . . . . . . . . 10 ⊢ ((π ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (π / 2) = ((1 / 2) · π))
27	14, 24, 25, 26	mp3an 1459	. . . . . . . . 9 ⊢ (π / 2) = ((1 / 2) · π)
28		6cn 12307	. . . . . . . . . 10 ⊢ 6 ∈ ℂ
29	18	nnne0i 12256	. . . . . . . . . 10 ⊢ 6 ≠ 0
30		divrec2 11893	. . . . . . . . . 10 ⊢ ((π ∈ ℂ ∧ 6 ∈ ℂ ∧ 6 ≠ 0) → (π / 6) = ((1 / 6) · π))
31	14, 28, 29, 30	mp3an 1459	. . . . . . . . 9 ⊢ (π / 6) = ((1 / 6) · π)
32	27, 31	oveq12i 7423	. . . . . . . 8 ⊢ ((π / 2) + (π / 6)) = (((1 / 2) · π) + ((1 / 6) · π))
33	24, 25	reccli 11948	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℂ
34	28, 29	reccli 11948	. . . . . . . . 9 ⊢ (1 / 6) ∈ ℂ
35	33, 34, 14	adddiri 11231	. . . . . . . 8 ⊢ (((1 / 2) + (1 / 6)) · π) = (((1 / 2) · π) + ((1 / 6) · π))
36		halfpm6th 12437	. . . . . . . . . 10 ⊢ (((1 / 2) − (1 / 6)) = (1 / 3) ∧ ((1 / 2) + (1 / 6)) = (2 / 3))
37	36	simpri 484	. . . . . . . . 9 ⊢ ((1 / 2) + (1 / 6)) = (2 / 3)
38	37	oveq1i 7421	. . . . . . . 8 ⊢ (((1 / 2) + (1 / 6)) · π) = ((2 / 3) · π)
39	32, 35, 38	3eqtr2i 2764	. . . . . . 7 ⊢ ((π / 2) + (π / 6)) = ((2 / 3) · π)
40	39	fveq2i 6893	. . . . . 6 ⊢ (cos‘((π / 2) + (π / 6))) = (cos‘((2 / 3) · π))
41		sincos6thpi 26261	. . . . . . . . 9 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))
42	41	simpli 482	. . . . . . . 8 ⊢ (sin‘(π / 6)) = (1 / 2)
43	42	negeqi 11457	. . . . . . 7 ⊢ -(sin‘(π / 6)) = -(1 / 2)
44		ax-1cn 11170	. . . . . . . 8 ⊢ 1 ∈ ℂ
45		divneg 11910	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(1 / 2) = (-1 / 2))
46	44, 24, 25, 45	mp3an 1459	. . . . . . 7 ⊢ -(1 / 2) = (-1 / 2)
47	43, 46	eqtri 2758	. . . . . 6 ⊢ -(sin‘(π / 6)) = (-1 / 2)
48	23, 40, 47	3eqtr3i 2766	. . . . 5 ⊢ (cos‘((2 / 3) · π)) = (-1 / 2)
49		sinhalfpip 26238	. . . . . . . . 9 ⊢ ((π / 6) ∈ ℂ → (sin‘((π / 2) + (π / 6))) = (cos‘(π / 6)))
50	21, 49	ax-mp 5	. . . . . . . 8 ⊢ (sin‘((π / 2) + (π / 6))) = (cos‘(π / 6))
51	39	fveq2i 6893	. . . . . . . 8 ⊢ (sin‘((π / 2) + (π / 6))) = (sin‘((2 / 3) · π))
52	41	simpri 484	. . . . . . . 8 ⊢ (cos‘(π / 6)) = ((√‘3) / 2)
53	50, 51, 52	3eqtr3i 2766	. . . . . . 7 ⊢ (sin‘((2 / 3) · π)) = ((√‘3) / 2)
54	53	oveq2i 7422	. . . . . 6 ⊢ (i · (sin‘((2 / 3) · π))) = (i · ((√‘3) / 2))
55		3re 12296	. . . . . . . . 9 ⊢ 3 ∈ ℝ
56		3nn0 12494	. . . . . . . . . 10 ⊢ 3 ∈ ℕ₀
57	56	nn0ge0i 12503	. . . . . . . . 9 ⊢ 0 ≤ 3
58		resqrtcl 15204	. . . . . . . . 9 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → (√‘3) ∈ ℝ)
59	55, 57, 58	mp2an 688	. . . . . . . 8 ⊢ (√‘3) ∈ ℝ
60	59	recni 11232	. . . . . . 7 ⊢ (√‘3) ∈ ℂ
61	12, 60, 24, 25	divassi 11974	. . . . . 6 ⊢ ((i · (√‘3)) / 2) = (i · ((√‘3) / 2))
62	54, 61	eqtr4i 2761	. . . . 5 ⊢ (i · (sin‘((2 / 3) · π))) = ((i · (√‘3)) / 2)
63	48, 62	oveq12i 7423	. . . 4 ⊢ ((cos‘((2 / 3) · π)) + (i · (sin‘((2 / 3) · π)))) = ((-1 / 2) + ((i · (√‘3)) / 2))
64	7, 14	mulcli 11225	. . . . 5 ⊢ ((2 / 3) · π) ∈ ℂ
65		efival 16099	. . . . 5 ⊢ (((2 / 3) · π) ∈ ℂ → (exp‘(i · ((2 / 3) · π))) = ((cos‘((2 / 3) · π)) + (i · (sin‘((2 / 3) · π)))))
66	64, 65	ax-mp 5	. . . 4 ⊢ (exp‘(i · ((2 / 3) · π))) = ((cos‘((2 / 3) · π)) + (i · (sin‘((2 / 3) · π))))
67	12, 60	mulcli 11225	. . . . 5 ⊢ (i · (√‘3)) ∈ ℂ
68	1, 67, 24, 25	divdiri 11975	. . . 4 ⊢ ((-1 + (i · (√‘3))) / 2) = ((-1 / 2) + ((i · (√‘3)) / 2))
69	63, 66, 68	3eqtr4i 2768	. . 3 ⊢ (exp‘(i · ((2 / 3) · π))) = ((-1 + (i · (√‘3))) / 2)
70	9, 17, 69	3eqtri 2762	. 2 ⊢ (-1↑_𝑐(2 / 3)) = ((-1 + (i · (√‘3))) / 2)
71		1z 12596	. . . 4 ⊢ 1 ∈ ℤ
72		root1cj 26500	. . . 4 ⊢ ((3 ∈ ℕ ∧ 1 ∈ ℤ) → (∗‘((-1↑_𝑐(2 / 3))↑1)) = ((-1↑_𝑐(2 / 3))↑(3 − 1)))
73	4, 71, 72	mp2an 688	. . 3 ⊢ (∗‘((-1↑_𝑐(2 / 3))↑1)) = ((-1↑_𝑐(2 / 3))↑(3 − 1))
74		cxpcl 26418	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ (2 / 3) ∈ ℂ) → (-1↑_𝑐(2 / 3)) ∈ ℂ)
75	1, 7, 74	mp2an 688	. . . . . . 7 ⊢ (-1↑_𝑐(2 / 3)) ∈ ℂ
76		exp1 14037	. . . . . . 7 ⊢ ((-1↑_𝑐(2 / 3)) ∈ ℂ → ((-1↑_𝑐(2 / 3))↑1) = (-1↑_𝑐(2 / 3)))
77	75, 76	ax-mp 5	. . . . . 6 ⊢ ((-1↑_𝑐(2 / 3))↑1) = (-1↑_𝑐(2 / 3))
78	77, 70	eqtri 2758	. . . . 5 ⊢ ((-1↑_𝑐(2 / 3))↑1) = ((-1 + (i · (√‘3))) / 2)
79	78	fveq2i 6893	. . . 4 ⊢ (∗‘((-1↑_𝑐(2 / 3))↑1)) = (∗‘((-1 + (i · (√‘3))) / 2))
80	1, 67	addcli 11224	. . . . . 6 ⊢ (-1 + (i · (√‘3))) ∈ ℂ
81	80, 24	cjdivi 15142	. . . . 5 ⊢ (2 ≠ 0 → (∗‘((-1 + (i · (√‘3))) / 2)) = ((∗‘(-1 + (i · (√‘3)))) / (∗‘2)))
82	25, 81	ax-mp 5	. . . 4 ⊢ (∗‘((-1 + (i · (√‘3))) / 2)) = ((∗‘(-1 + (i · (√‘3)))) / (∗‘2))
83	1, 67	cjaddi 15139	. . . . . 6 ⊢ (∗‘(-1 + (i · (√‘3)))) = ((∗‘-1) + (∗‘(i · (√‘3))))
84		neg1rr 12331	. . . . . . . 8 ⊢ -1 ∈ ℝ
85		cjre 15090	. . . . . . . 8 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1)
86	84, 85	ax-mp 5	. . . . . . 7 ⊢ (∗‘-1) = -1
87	12, 60	cjmuli 15140	. . . . . . . 8 ⊢ (∗‘(i · (√‘3))) = ((∗‘i) · (∗‘(√‘3)))
88		cji 15110	. . . . . . . . 9 ⊢ (∗‘i) = -i
89		cjre 15090	. . . . . . . . . 10 ⊢ ((√‘3) ∈ ℝ → (∗‘(√‘3)) = (√‘3))
90	59, 89	ax-mp 5	. . . . . . . . 9 ⊢ (∗‘(√‘3)) = (√‘3)
91	88, 90	oveq12i 7423	. . . . . . . 8 ⊢ ((∗‘i) · (∗‘(√‘3))) = (-i · (√‘3))
92	12, 60	mulneg1i 11664	. . . . . . . 8 ⊢ (-i · (√‘3)) = -(i · (√‘3))
93	87, 91, 92	3eqtri 2762	. . . . . . 7 ⊢ (∗‘(i · (√‘3))) = -(i · (√‘3))
94	86, 93	oveq12i 7423	. . . . . 6 ⊢ ((∗‘-1) + (∗‘(i · (√‘3)))) = (-1 + -(i · (√‘3)))
95	1, 67	negsubi 11542	. . . . . 6 ⊢ (-1 + -(i · (√‘3))) = (-1 − (i · (√‘3)))
96	83, 94, 95	3eqtri 2762	. . . . 5 ⊢ (∗‘(-1 + (i · (√‘3)))) = (-1 − (i · (√‘3)))
97		cjre 15090	. . . . . 6 ⊢ (2 ∈ ℝ → (∗‘2) = 2)
98	3, 97	ax-mp 5	. . . . 5 ⊢ (∗‘2) = 2
99	96, 98	oveq12i 7423	. . . 4 ⊢ ((∗‘(-1 + (i · (√‘3)))) / (∗‘2)) = ((-1 − (i · (√‘3))) / 2)
100	79, 82, 99	3eqtri 2762	. . 3 ⊢ (∗‘((-1↑_𝑐(2 / 3))↑1)) = ((-1 − (i · (√‘3))) / 2)
101		3m1e2 12344	. . . 4 ⊢ (3 − 1) = 2
102	101	oveq2i 7422	. . 3 ⊢ ((-1↑_𝑐(2 / 3))↑(3 − 1)) = ((-1↑_𝑐(2 / 3))↑2)
103	73, 100, 102	3eqtr3ri 2767	. 2 ⊢ ((-1↑_𝑐(2 / 3))↑2) = ((-1 − (i · (√‘3))) / 2)
104	70, 103	pm3.2i 469	1 ⊢ ((-1↑_𝑐(2 / 3)) = ((-1 + (i · (√‘3))) / 2) ∧ ((-1↑_𝑐(2 / 3))↑2) = ((-1 − (i · (√‘3))) / 2))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 ici 11114 + caddc 11115 · cmul 11117 ≤ cle 11253 − cmin 11448 -cneg 11449 / cdiv 11875 ℕcn 12216 2c2 12271 3c3 12272 6c6 12275 ℤcz 12562 ↑cexp 14031 ∗ccj 15047 √csqrt 15184 expce 16009 sincsin 16011 cosccos 16012 πcpi 16014 logclog 26299 ↑_𝑐ccxp 26300

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-limc 25615 df-dv 25616 df-log 26301 df-cxp 26302

This theorem is referenced by: 1cubr 26583

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