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Theorem cubic 25027

Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 4473 to convert the existential quantifier to a triple disjunction. This is Metamath 100 proof #37. (Contributed by Mario Carneiro, 26-Apr-2015.)

**Hypotheses**
Ref	Expression
cubic.r	⊢ 𝑅 = {1, ((-1 + (i · (√‘3))) / 2), ((-1 − (i · (√‘3))) / 2)}
cubic.a	⊢ (𝜑 → 𝐴 ∈ ℂ)
cubic.z	⊢ (𝜑 → 𝐴 ≠ 0)
cubic.b	⊢ (𝜑 → 𝐵 ∈ ℂ)
cubic.c	⊢ (𝜑 → 𝐶 ∈ ℂ)
cubic.d	⊢ (𝜑 → 𝐷 ∈ ℂ)
cubic.x	⊢ (𝜑 → 𝑋 ∈ ℂ)
cubic.t	⊢ (𝜑 → 𝑇 = (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)))
cubic.g	⊢ (𝜑 → 𝐺 = ((𝑁↑2) − (4 · (𝑀↑3))))
cubic.m	⊢ (𝜑 → 𝑀 = ((𝐵↑2) − (3 · (𝐴 · 𝐶))))
cubic.n	⊢ (𝜑 → 𝑁 = (((2 · (𝐵↑3)) − ((9 · 𝐴) · (𝐵 · 𝐶))) + (;27 · ((𝐴↑2) · 𝐷))))
cubic.0	⊢ (𝜑 → 𝑀 ≠ 0)

**Assertion**
Ref	Expression
cubic	⊢ (𝜑 → ((((𝐴 · (𝑋↑3)) + (𝐵 · (𝑋↑2))) + ((𝐶 · 𝑋) + 𝐷)) = 0 ↔ ∃𝑟 ∈ 𝑅 𝑋 = -(((𝐵 + (𝑟 · 𝑇)) + (𝑀 / (𝑟 · 𝑇))) / (3 · 𝐴))))

Distinct variable groups: 𝐴,𝑟 𝐵,𝑟 𝑀,𝑟 𝑁,𝑟 𝜑,𝑟 𝑇,𝑟 𝑋,𝑟 Allowed substitution hints: 𝐶(𝑟) 𝐷(𝑟) 𝑅(𝑟) 𝐺(𝑟)

**Proof of Theorem cubic**
Step	Hyp	Ref	Expression
1		cubic.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		cubic.z	. . 3 ⊢ (𝜑 → 𝐴 ≠ 0)
3		cubic.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ)
4		cubic.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5		cubic.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ)
6		cubic.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ)
7		cubic.t	. . . 4 ⊢ (𝜑 → 𝑇 = (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)))
8		cubic.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (((2 · (𝐵↑3)) − ((9 · 𝐴) · (𝐵 · 𝐶))) + (;27 · ((𝐴↑2) · 𝐷))))
9		2cn 11450	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
10		3nn0 11662	. . . . . . . . . . . 12 ⊢ 3 ∈ ℕ₀
11		expcl 13196	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐵↑3) ∈ ℂ)
12	3, 10, 11	sylancl 580	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵↑3) ∈ ℂ)
13		mulcl 10356	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ (𝐵↑3) ∈ ℂ) → (2 · (𝐵↑3)) ∈ ℂ)
14	9, 12, 13	sylancr 581	. . . . . . . . . 10 ⊢ (𝜑 → (2 · (𝐵↑3)) ∈ ℂ)
15		9cn 11481	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
16		mulcl 10356	. . . . . . . . . . . 12 ⊢ ((9 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (9 · 𝐴) ∈ ℂ)
17	15, 1, 16	sylancr 581	. . . . . . . . . . 11 ⊢ (𝜑 → (9 · 𝐴) ∈ ℂ)
18	3, 4	mulcld 10397	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 · 𝐶) ∈ ℂ)
19	17, 18	mulcld 10397	. . . . . . . . . 10 ⊢ (𝜑 → ((9 · 𝐴) · (𝐵 · 𝐶)) ∈ ℂ)
20	14, 19	subcld 10734	. . . . . . . . 9 ⊢ (𝜑 → ((2 · (𝐵↑3)) − ((9 · 𝐴) · (𝐵 · 𝐶))) ∈ ℂ)
21		2nn0 11661	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
22		7nn 11471	. . . . . . . . . . . 12 ⊢ 7 ∈ ℕ
23	21, 22	decnncl 11866	. . . . . . . . . . 11 ⊢ ;27 ∈ ℕ
24	23	nncni 11385	. . . . . . . . . 10 ⊢ ;27 ∈ ℂ
25	1	sqcld 13325	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ)
26	25, 5	mulcld 10397	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴↑2) · 𝐷) ∈ ℂ)
27		mulcl 10356	. . . . . . . . . 10 ⊢ ((;27 ∈ ℂ ∧ ((𝐴↑2) · 𝐷) ∈ ℂ) → (;27 · ((𝐴↑2) · 𝐷)) ∈ ℂ)
28	24, 26, 27	sylancr 581	. . . . . . . . 9 ⊢ (𝜑 → (;27 · ((𝐴↑2) · 𝐷)) ∈ ℂ)
29	20, 28	addcld 10396	. . . . . . . 8 ⊢ (𝜑 → (((2 · (𝐵↑3)) − ((9 · 𝐴) · (𝐵 · 𝐶))) + (;27 · ((𝐴↑2) · 𝐷))) ∈ ℂ)
30	8, 29	eqeltrd 2859	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
31		cubic.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((𝑁↑2) − (4 · (𝑀↑3))))
32	30	sqcld 13325	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ)
33		4cn 11461	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
34		cubic.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 = ((𝐵↑2) − (3 · (𝐴 · 𝐶))))
35	3	sqcld 13325	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ)
36		3cn 11456	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ ℂ
37	1, 4	mulcld 10397	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℂ)
38		mulcl 10356	. . . . . . . . . . . . . . 15 ⊢ ((3 ∈ ℂ ∧ (𝐴 · 𝐶) ∈ ℂ) → (3 · (𝐴 · 𝐶)) ∈ ℂ)
39	36, 37, 38	sylancr 581	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (3 · (𝐴 · 𝐶)) ∈ ℂ)
40	35, 39	subcld 10734	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵↑2) − (3 · (𝐴 · 𝐶))) ∈ ℂ)
41	34, 40	eqeltrd 2859	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ)
42		expcl 13196	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑀↑3) ∈ ℂ)
43	41, 10, 42	sylancl 580	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀↑3) ∈ ℂ)
44		mulcl 10356	. . . . . . . . . . 11 ⊢ ((4 ∈ ℂ ∧ (𝑀↑3) ∈ ℂ) → (4 · (𝑀↑3)) ∈ ℂ)
45	33, 43, 44	sylancr 581	. . . . . . . . . 10 ⊢ (𝜑 → (4 · (𝑀↑3)) ∈ ℂ)
46	32, 45	subcld 10734	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − (4 · (𝑀↑3))) ∈ ℂ)
47	31, 46	eqeltrd 2859	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℂ)
48	47	sqrtcld 14584	. . . . . . 7 ⊢ (𝜑 → (√‘𝐺) ∈ ℂ)
49	30, 48	addcld 10396	. . . . . 6 ⊢ (𝜑 → (𝑁 + (√‘𝐺)) ∈ ℂ)
50	49	halfcld 11627	. . . . 5 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) / 2) ∈ ℂ)
51		3ne0 11488	. . . . . 6 ⊢ 3 ≠ 0
52	36, 51	reccli 11105	. . . . 5 ⊢ (1 / 3) ∈ ℂ
53		cxpcl 24857	. . . . 5 ⊢ ((((𝑁 + (√‘𝐺)) / 2) ∈ ℂ ∧ (1 / 3) ∈ ℂ) → (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)) ∈ ℂ)
54	50, 52, 53	sylancl 580	. . . 4 ⊢ (𝜑 → (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)) ∈ ℂ)
55	7, 54	eqeltrd 2859	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
56	7	oveq1d 6937	. . . 4 ⊢ (𝜑 → (𝑇↑3) = ((((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3))↑3))
57		3nn 11454	. . . . 5 ⊢ 3 ∈ ℕ
58		cxproot 24873	. . . . 5 ⊢ ((((𝑁 + (√‘𝐺)) / 2) ∈ ℂ ∧ 3 ∈ ℕ) → ((((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3))↑3) = ((𝑁 + (√‘𝐺)) / 2))
59	50, 57, 58	sylancl 580	. . . 4 ⊢ (𝜑 → ((((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3))↑3) = ((𝑁 + (√‘𝐺)) / 2))
60	56, 59	eqtrd 2814	. . 3 ⊢ (𝜑 → (𝑇↑3) = ((𝑁 + (√‘𝐺)) / 2))
61	47	sqsqrtd 14586	. . . 4 ⊢ (𝜑 → ((√‘𝐺)↑2) = 𝐺)
62	61, 31	eqtrd 2814	. . 3 ⊢ (𝜑 → ((√‘𝐺)↑2) = ((𝑁↑2) − (4 · (𝑀↑3))))
63	9	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
64	33	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 4 ∈ ℂ)
65		4ne0 11490	. . . . . . . . . 10 ⊢ 4 ≠ 0
66	65	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 4 ≠ 0)
67		cubic.0	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≠ 0)
68		3z 11762	. . . . . . . . . . 11 ⊢ 3 ∈ ℤ
69	68	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℤ)
70	41, 67, 69	expne0d 13333	. . . . . . . . 9 ⊢ (𝜑 → (𝑀↑3) ≠ 0)
71	64, 43, 66, 70	mulne0d 11027	. . . . . . . 8 ⊢ (𝜑 → (4 · (𝑀↑3)) ≠ 0)
72	62	oveq2d 6938	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((√‘𝐺)↑2)) = ((𝑁↑2) − ((𝑁↑2) − (4 · (𝑀↑3)))))
73		subsq 13291	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ (√‘𝐺) ∈ ℂ) → ((𝑁↑2) − ((√‘𝐺)↑2)) = ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))))
74	30, 48, 73	syl2anc 579	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((√‘𝐺)↑2)) = ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))))
75	32, 45	nncand 10739	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((𝑁↑2) − (4 · (𝑀↑3)))) = (4 · (𝑀↑3)))
76	72, 74, 75	3eqtr3d 2822	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) = (4 · (𝑀↑3)))
77	30, 48	subcld 10734	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − (√‘𝐺)) ∈ ℂ)
78	77	mul02d 10574	. . . . . . . 8 ⊢ (𝜑 → (0 · (𝑁 − (√‘𝐺))) = 0)
79	71, 76, 78	3netr4d 3046	. . . . . . 7 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) ≠ (0 · (𝑁 − (√‘𝐺))))
80		oveq1 6929	. . . . . . . 8 ⊢ ((𝑁 + (√‘𝐺)) = 0 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) = (0 · (𝑁 − (√‘𝐺))))
81	80	necon3i 3001	. . . . . . 7 ⊢ (((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) ≠ (0 · (𝑁 − (√‘𝐺))) → (𝑁 + (√‘𝐺)) ≠ 0)
82	79, 81	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 + (√‘𝐺)) ≠ 0)
83		2ne0 11486	. . . . . . 7 ⊢ 2 ≠ 0
84	83	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
85	49, 63, 82, 84	divne0d 11167	. . . . 5 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) / 2) ≠ 0)
86	52	a1i 11	. . . . 5 ⊢ (𝜑 → (1 / 3) ∈ ℂ)
87	50, 85, 86	cxpne0d 24896	. . . 4 ⊢ (𝜑 → (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)) ≠ 0)
88	7, 87	eqnetrd 3036	. . 3 ⊢ (𝜑 → 𝑇 ≠ 0)
89	1, 2, 3, 4, 5, 6, 55, 60, 48, 62, 34, 8, 88	cubic2 25026	. 2 ⊢ (𝜑 → ((((𝐴 · (𝑋↑3)) + (𝐵 · (𝑋↑2))) + ((𝐶 · 𝑋) + 𝐷)) = 0 ↔ ∃𝑟 ∈ ℂ ((𝑟↑3) = 1 ∧ 𝑋 = -(((𝐵 + (𝑟 · 𝑇)) + (𝑀 / (𝑟 · 𝑇))) / (3 · 𝐴)))))
90		cubic.r	. . . . . 6 ⊢ 𝑅 = {1, ((-1 + (i · (√‘3))) / 2), ((-1 − (i · (√‘3))) / 2)}
91	90	1cubr 25020	. . . . 5 ⊢ (𝑟 ∈ 𝑅 ↔ (𝑟 ∈ ℂ ∧ (𝑟↑3) = 1))
92	91	anbi1i 617	. . . 4 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑋 = -(((𝐵 + (𝑟 · 𝑇)) + (𝑀 / (𝑟 · 𝑇))) / (3 · 𝐴))) ↔ ((𝑟 ∈ ℂ ∧ (𝑟↑3) = 1) ∧ 𝑋 = -(((𝐵 + (𝑟 · 𝑇)) + (𝑀 / (𝑟 · 𝑇))) / (3 · 𝐴))))
93		anass 462	. . . 4 ⊢ (((𝑟 ∈ ℂ ∧ (𝑟↑3) = 1) ∧ 𝑋 = -(((𝐵 + (𝑟 · 𝑇)) + (𝑀 / (𝑟 · 𝑇))) / (3 · 𝐴))) ↔ (𝑟 ∈ ℂ ∧ ((𝑟↑3) = 1 ∧ 𝑋 = -(((𝐵 + (𝑟 · 𝑇)) + (𝑀 / (𝑟 · 𝑇))) / (3 · 𝐴)))))
94	92, 93	bitri 267	. . 3 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑋 = -(((𝐵 + (𝑟 · 𝑇)) + (𝑀 / (𝑟 · 𝑇))) / (3 · 𝐴))) ↔ (𝑟 ∈ ℂ ∧ ((𝑟↑3) = 1 ∧ 𝑋 = -(((𝐵 + (𝑟 · 𝑇)) + (𝑀 / (𝑟 · 𝑇))) / (3 · 𝐴)))))
95	94	rexbii2 3222	. 2 ⊢ (∃𝑟 ∈ 𝑅 𝑋 = -(((𝐵 + (𝑟 · 𝑇)) + (𝑀 / (𝑟 · 𝑇))) / (3 · 𝐴)) ↔ ∃𝑟 ∈ ℂ ((𝑟↑3) = 1 ∧ 𝑋 = -(((𝐵 + (𝑟 · 𝑇)) + (𝑀 / (𝑟 · 𝑇))) / (3 · 𝐴))))
96	89, 95	syl6bbr 281	1 ⊢ (𝜑 → ((((𝐴 · (𝑋↑3)) + (𝐵 · (𝑋↑2))) + ((𝐶 · 𝑋) + 𝐷)) = 0 ↔ ∃𝑟 ∈ 𝑅 𝑋 = -(((𝐵 + (𝑟 · 𝑇)) + (𝑀 / (𝑟 · 𝑇))) / (3 · 𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∃wrex 3091 {ctp 4402 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 0cc0 10272 1c1 10273 ici 10274 + caddc 10275 · cmul 10277 − cmin 10606 -cneg 10607 / cdiv 11032 ℕcn 11374 2c2 11430 3c3 11431 4c4 11432 7c7 11435 9c9 11437 ℕ₀cn0 11642 ℤcz 11728 cdc 11845 ↑cexp 13178 √csqrt 14380 ↑_𝑐ccxp 24739

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-fac 13379 df-bc 13408 df-hash 13436 df-shft 14214 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-sum 14825 df-ef 15200 df-sin 15202 df-cos 15203 df-pi 15205 df-dvds 15388 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cn 21439 df-cnp 21440 df-haus 21527 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cncf 23089 df-limc 24067 df-dv 24068 df-log 24740 df-cxp 24741

This theorem is referenced by: (None)

W3C validator