1cubrlem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 1cubrlem		Structured version Visualization version GIF version

Theorem 1cubrlem 26346

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 12326	. . . 4 ⊢ -1 ∈ ℂ
2		neg1ne0 12328	. . . 4 ⊢ -1 ≠ 0
3		2re 12286	. . . . . 6 ⊢ 2 ∈ ℝ
4		3nn 12291	. . . . . 6 ⊢ 3 ∈ ℕ
5		nndivre 12253	. . . . . 6 ⊢ ((2 ∈ ℝ ∧ 3 ∈ ℕ) → (2 / 3) ∈ ℝ)
6	3, 4, 5	mp2an 691	. . . . 5 ⊢ (2 / 3) ∈ ℝ
7	6	recni 11228	. . . 4 ⊢ (2 / 3) ∈ ℂ
8		cxpef 26173	. . . 4 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0 ∧ (2 / 3) ∈ ℂ) → (-1↑_𝑐(2 / 3)) = (exp‘((2 / 3) · (log‘-1))))
9	1, 2, 7, 8	mp3an 1462	. . 3 ⊢ (-1↑_𝑐(2 / 3)) = (exp‘((2 / 3) · (log‘-1)))
10		logm1 26097	. . . . . 6 ⊢ (log‘-1) = (i · π)
11	10	oveq2i 7420	. . . . 5 ⊢ ((2 / 3) · (log‘-1)) = ((2 / 3) · (i · π))
12		ax-icn 11169	. . . . . 6 ⊢ i ∈ ℂ
13		pire 25968	. . . . . . 7 ⊢ π ∈ ℝ
14	13	recni 11228	. . . . . 6 ⊢ π ∈ ℂ
15	7, 12, 14	mul12i 11409	. . . . 5 ⊢ ((2 / 3) · (i · π)) = (i · ((2 / 3) · π))
16	11, 15	eqtri 2761	. . . 4 ⊢ ((2 / 3) · (log‘-1)) = (i · ((2 / 3) · π))
17	16	fveq2i 6895	. . 3 ⊢ (exp‘((2 / 3) · (log‘-1))) = (exp‘(i · ((2 / 3) · π)))
18		6nn 12301	. . . . . . . . 9 ⊢ 6 ∈ ℕ
19		nndivre 12253	. . . . . . . . 9 ⊢ ((π ∈ ℝ ∧ 6 ∈ ℕ) → (π / 6) ∈ ℝ)
20	13, 18, 19	mp2an 691	. . . . . . . 8 ⊢ (π / 6) ∈ ℝ
21	20	recni 11228	. . . . . . 7 ⊢ (π / 6) ∈ ℂ
22		coshalfpip 26004	. . . . . . 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 12287	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
25		2ne0 12316	. . . . . . . . . 10 ⊢ 2 ≠ 0
26		divrec2 11889	. . . . . . . . . 10 ⊢ ((π ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (π / 2) = ((1 / 2) · π))
27	14, 24, 25, 26	mp3an 1462	. . . . . . . . 9 ⊢ (π / 2) = ((1 / 2) · π)
28		6cn 12303	. . . . . . . . . 10 ⊢ 6 ∈ ℂ
29	18	nnne0i 12252	. . . . . . . . . 10 ⊢ 6 ≠ 0
30		divrec2 11889	. . . . . . . . . 10 ⊢ ((π ∈ ℂ ∧ 6 ∈ ℂ ∧ 6 ≠ 0) → (π / 6) = ((1 / 6) · π))
31	14, 28, 29, 30	mp3an 1462	. . . . . . . . 9 ⊢ (π / 6) = ((1 / 6) · π)
32	27, 31	oveq12i 7421	. . . . . . . 8 ⊢ ((π / 2) + (π / 6)) = (((1 / 2) · π) + ((1 / 6) · π))
33	24, 25	reccli 11944	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℂ
34	28, 29	reccli 11944	. . . . . . . . 9 ⊢ (1 / 6) ∈ ℂ
35	33, 34, 14	adddiri 11227	. . . . . . . 8 ⊢ (((1 / 2) + (1 / 6)) · π) = (((1 / 2) · π) + ((1 / 6) · π))
36		halfpm6th 12433	. . . . . . . . . 10 ⊢ (((1 / 2) − (1 / 6)) = (1 / 3) ∧ ((1 / 2) + (1 / 6)) = (2 / 3))
37	36	simpri 487	. . . . . . . . 9 ⊢ ((1 / 2) + (1 / 6)) = (2 / 3)
38	37	oveq1i 7419	. . . . . . . 8 ⊢ (((1 / 2) + (1 / 6)) · π) = ((2 / 3) · π)
39	32, 35, 38	3eqtr2i 2767	. . . . . . 7 ⊢ ((π / 2) + (π / 6)) = ((2 / 3) · π)
40	39	fveq2i 6895	. . . . . 6 ⊢ (cos‘((π / 2) + (π / 6))) = (cos‘((2 / 3) · π))
41		sincos6thpi 26025	. . . . . . . . 9 ⊢ ((sin‘(π / 6)) = (1 / 2) ∧ (cos‘(π / 6)) = ((√‘3) / 2))
42	41	simpli 485	. . . . . . . 8 ⊢ (sin‘(π / 6)) = (1 / 2)
43	42	negeqi 11453	. . . . . . 7 ⊢ -(sin‘(π / 6)) = -(1 / 2)
44		ax-1cn 11168	. . . . . . . 8 ⊢ 1 ∈ ℂ
45		divneg 11906	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(1 / 2) = (-1 / 2))
46	44, 24, 25, 45	mp3an 1462	. . . . . . 7 ⊢ -(1 / 2) = (-1 / 2)
47	43, 46	eqtri 2761	. . . . . 6 ⊢ -(sin‘(π / 6)) = (-1 / 2)
48	23, 40, 47	3eqtr3i 2769	. . . . 5 ⊢ (cos‘((2 / 3) · π)) = (-1 / 2)
49		sinhalfpip 26002	. . . . . . . . 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 6895	. . . . . . . 8 ⊢ (sin‘((π / 2) + (π / 6))) = (sin‘((2 / 3) · π))
52	41	simpri 487	. . . . . . . 8 ⊢ (cos‘(π / 6)) = ((√‘3) / 2)
53	50, 51, 52	3eqtr3i 2769	. . . . . . 7 ⊢ (sin‘((2 / 3) · π)) = ((√‘3) / 2)
54	53	oveq2i 7420	. . . . . 6 ⊢ (i · (sin‘((2 / 3) · π))) = (i · ((√‘3) / 2))
55		3re 12292	. . . . . . . . 9 ⊢ 3 ∈ ℝ
56		3nn0 12490	. . . . . . . . . 10 ⊢ 3 ∈ ℕ₀
57	56	nn0ge0i 12499	. . . . . . . . 9 ⊢ 0 ≤ 3
58		resqrtcl 15200	. . . . . . . . 9 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → (√‘3) ∈ ℝ)
59	55, 57, 58	mp2an 691	. . . . . . . 8 ⊢ (√‘3) ∈ ℝ
60	59	recni 11228	. . . . . . 7 ⊢ (√‘3) ∈ ℂ
61	12, 60, 24, 25	divassi 11970	. . . . . 6 ⊢ ((i · (√‘3)) / 2) = (i · ((√‘3) / 2))
62	54, 61	eqtr4i 2764	. . . . 5 ⊢ (i · (sin‘((2 / 3) · π))) = ((i · (√‘3)) / 2)
63	48, 62	oveq12i 7421	. . . 4 ⊢ ((cos‘((2 / 3) · π)) + (i · (sin‘((2 / 3) · π)))) = ((-1 / 2) + ((i · (√‘3)) / 2))
64	7, 14	mulcli 11221	. . . . 5 ⊢ ((2 / 3) · π) ∈ ℂ
65		efival 16095	. . . . 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 11221	. . . . 5 ⊢ (i · (√‘3)) ∈ ℂ
68	1, 67, 24, 25	divdiri 11971	. . . 4 ⊢ ((-1 + (i · (√‘3))) / 2) = ((-1 / 2) + ((i · (√‘3)) / 2))
69	63, 66, 68	3eqtr4i 2771	. . 3 ⊢ (exp‘(i · ((2 / 3) · π))) = ((-1 + (i · (√‘3))) / 2)
70	9, 17, 69	3eqtri 2765	. 2 ⊢ (-1↑_𝑐(2 / 3)) = ((-1 + (i · (√‘3))) / 2)
71		1z 12592	. . . 4 ⊢ 1 ∈ ℤ
72		root1cj 26264	. . . 4 ⊢ ((3 ∈ ℕ ∧ 1 ∈ ℤ) → (∗‘((-1↑_𝑐(2 / 3))↑1)) = ((-1↑_𝑐(2 / 3))↑(3 − 1)))
73	4, 71, 72	mp2an 691	. . 3 ⊢ (∗‘((-1↑_𝑐(2 / 3))↑1)) = ((-1↑_𝑐(2 / 3))↑(3 − 1))
74		cxpcl 26182	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ (2 / 3) ∈ ℂ) → (-1↑_𝑐(2 / 3)) ∈ ℂ)
75	1, 7, 74	mp2an 691	. . . . . . 7 ⊢ (-1↑_𝑐(2 / 3)) ∈ ℂ
76		exp1 14033	. . . . . . 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 2761	. . . . 5 ⊢ ((-1↑_𝑐(2 / 3))↑1) = ((-1 + (i · (√‘3))) / 2)
79	78	fveq2i 6895	. . . 4 ⊢ (∗‘((-1↑_𝑐(2 / 3))↑1)) = (∗‘((-1 + (i · (√‘3))) / 2))
80	1, 67	addcli 11220	. . . . . 6 ⊢ (-1 + (i · (√‘3))) ∈ ℂ
81	80, 24	cjdivi 15138	. . . . 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 15135	. . . . . 6 ⊢ (∗‘(-1 + (i · (√‘3)))) = ((∗‘-1) + (∗‘(i · (√‘3))))
84		neg1rr 12327	. . . . . . . 8 ⊢ -1 ∈ ℝ
85		cjre 15086	. . . . . . . 8 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1)
86	84, 85	ax-mp 5	. . . . . . 7 ⊢ (∗‘-1) = -1
87	12, 60	cjmuli 15136	. . . . . . . 8 ⊢ (∗‘(i · (√‘3))) = ((∗‘i) · (∗‘(√‘3)))
88		cji 15106	. . . . . . . . 9 ⊢ (∗‘i) = -i
89		cjre 15086	. . . . . . . . . 10 ⊢ ((√‘3) ∈ ℝ → (∗‘(√‘3)) = (√‘3))
90	59, 89	ax-mp 5	. . . . . . . . 9 ⊢ (∗‘(√‘3)) = (√‘3)
91	88, 90	oveq12i 7421	. . . . . . . 8 ⊢ ((∗‘i) · (∗‘(√‘3))) = (-i · (√‘3))
92	12, 60	mulneg1i 11660	. . . . . . . 8 ⊢ (-i · (√‘3)) = -(i · (√‘3))
93	87, 91, 92	3eqtri 2765	. . . . . . 7 ⊢ (∗‘(i · (√‘3))) = -(i · (√‘3))
94	86, 93	oveq12i 7421	. . . . . 6 ⊢ ((∗‘-1) + (∗‘(i · (√‘3)))) = (-1 + -(i · (√‘3)))
95	1, 67	negsubi 11538	. . . . . 6 ⊢ (-1 + -(i · (√‘3))) = (-1 − (i · (√‘3)))
96	83, 94, 95	3eqtri 2765	. . . . 5 ⊢ (∗‘(-1 + (i · (√‘3)))) = (-1 − (i · (√‘3)))
97		cjre 15086	. . . . . 6 ⊢ (2 ∈ ℝ → (∗‘2) = 2)
98	3, 97	ax-mp 5	. . . . 5 ⊢ (∗‘2) = 2
99	96, 98	oveq12i 7421	. . . 4 ⊢ ((∗‘(-1 + (i · (√‘3)))) / (∗‘2)) = ((-1 − (i · (√‘3))) / 2)
100	79, 82, 99	3eqtri 2765	. . 3 ⊢ (∗‘((-1↑_𝑐(2 / 3))↑1)) = ((-1 − (i · (√‘3))) / 2)
101		3m1e2 12340	. . . 4 ⊢ (3 − 1) = 2
102	101	oveq2i 7420	. . 3 ⊢ ((-1↑_𝑐(2 / 3))↑(3 − 1)) = ((-1↑_𝑐(2 / 3))↑2)
103	73, 100, 102	3eqtr3ri 2770	. 2 ⊢ ((-1↑_𝑐(2 / 3))↑2) = ((-1 − (i · (√‘3))) / 2)
104	70, 103	pm3.2i 472	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 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 ici 11112 + caddc 11113 · cmul 11115 ≤ cle 11249 − cmin 11444 -cneg 11445 / cdiv 11871 ℕcn 12212 2c2 12267 3c3 12268 6c6 12271 ℤcz 12558 ↑cexp 14027 ∗ccj 15043 √csqrt 15180 expce 16005 sincsin 16007 cosccos 16008 πcpi 16010 logclog 26063 ↑_𝑐ccxp 26064

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-cxp 26066

This theorem is referenced by: 1cubr 26347

W3C validator