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Theorem cubic 26697

Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 4702 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 12284	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
10		3nn0 12487	. . . . . . . . . . . 12 ⊢ 3 ∈ ℕ₀
11		expcl 14042	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐵↑3) ∈ ℂ)
12	3, 10, 11	sylancl 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵↑3) ∈ ℂ)
13		mulcl 11190	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ (𝐵↑3) ∈ ℂ) → (2 · (𝐵↑3)) ∈ ℂ)
14	9, 12, 13	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → (2 · (𝐵↑3)) ∈ ℂ)
15		9cn 12309	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
16		mulcl 11190	. . . . . . . . . . . 12 ⊢ ((9 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (9 · 𝐴) ∈ ℂ)
17	15, 1, 16	sylancr 586	. . . . . . . . . . 11 ⊢ (𝜑 → (9 · 𝐴) ∈ ℂ)
18	3, 4	mulcld 11231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 · 𝐶) ∈ ℂ)
19	17, 18	mulcld 11231	. . . . . . . . . 10 ⊢ (𝜑 → ((9 · 𝐴) · (𝐵 · 𝐶)) ∈ ℂ)
20	14, 19	subcld 11568	. . . . . . . . 9 ⊢ (𝜑 → ((2 · (𝐵↑3)) − ((9 · 𝐴) · (𝐵 · 𝐶))) ∈ ℂ)
21		2nn0 12486	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
22		7nn 12301	. . . . . . . . . . . 12 ⊢ 7 ∈ ℕ
23	21, 22	decnncl 12694	. . . . . . . . . . 11 ⊢ ;27 ∈ ℕ
24	23	nncni 12219	. . . . . . . . . 10 ⊢ ;27 ∈ ℂ
25	1	sqcld 14106	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ)
26	25, 5	mulcld 11231	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴↑2) · 𝐷) ∈ ℂ)
27		mulcl 11190	. . . . . . . . . 10 ⊢ ((;27 ∈ ℂ ∧ ((𝐴↑2) · 𝐷) ∈ ℂ) → (;27 · ((𝐴↑2) · 𝐷)) ∈ ℂ)
28	24, 26, 27	sylancr 586	. . . . . . . . 9 ⊢ (𝜑 → (;27 · ((𝐴↑2) · 𝐷)) ∈ ℂ)
29	20, 28	addcld 11230	. . . . . . . 8 ⊢ (𝜑 → (((2 · (𝐵↑3)) − ((9 · 𝐴) · (𝐵 · 𝐶))) + (;27 · ((𝐴↑2) · 𝐷))) ∈ ℂ)
30	8, 29	eqeltrd 2825	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
31		cubic.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((𝑁↑2) − (4 · (𝑀↑3))))
32	30	sqcld 14106	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ)
33		4cn 12294	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
34		cubic.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 = ((𝐵↑2) − (3 · (𝐴 · 𝐶))))
35	3	sqcld 14106	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ)
36		3cn 12290	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ ℂ
37	1, 4	mulcld 11231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℂ)
38		mulcl 11190	. . . . . . . . . . . . . . 15 ⊢ ((3 ∈ ℂ ∧ (𝐴 · 𝐶) ∈ ℂ) → (3 · (𝐴 · 𝐶)) ∈ ℂ)
39	36, 37, 38	sylancr 586	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (3 · (𝐴 · 𝐶)) ∈ ℂ)
40	35, 39	subcld 11568	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵↑2) − (3 · (𝐴 · 𝐶))) ∈ ℂ)
41	34, 40	eqeltrd 2825	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ)
42		expcl 14042	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑀↑3) ∈ ℂ)
43	41, 10, 42	sylancl 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀↑3) ∈ ℂ)
44		mulcl 11190	. . . . . . . . . . 11 ⊢ ((4 ∈ ℂ ∧ (𝑀↑3) ∈ ℂ) → (4 · (𝑀↑3)) ∈ ℂ)
45	33, 43, 44	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → (4 · (𝑀↑3)) ∈ ℂ)
46	32, 45	subcld 11568	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − (4 · (𝑀↑3))) ∈ ℂ)
47	31, 46	eqeltrd 2825	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℂ)
48	47	sqrtcld 15381	. . . . . . 7 ⊢ (𝜑 → (√‘𝐺) ∈ ℂ)
49	30, 48	addcld 11230	. . . . . 6 ⊢ (𝜑 → (𝑁 + (√‘𝐺)) ∈ ℂ)
50	49	halfcld 12454	. . . . 5 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) / 2) ∈ ℂ)
51		3ne0 12315	. . . . . 6 ⊢ 3 ≠ 0
52	36, 51	reccli 11941	. . . . 5 ⊢ (1 / 3) ∈ ℂ
53		cxpcl 26524	. . . . 5 ⊢ ((((𝑁 + (√‘𝐺)) / 2) ∈ ℂ ∧ (1 / 3) ∈ ℂ) → (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)) ∈ ℂ)
54	50, 52, 53	sylancl 585	. . . 4 ⊢ (𝜑 → (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)) ∈ ℂ)
55	7, 54	eqeltrd 2825	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
56	7	oveq1d 7416	. . . 4 ⊢ (𝜑 → (𝑇↑3) = ((((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3))↑3))
57		3nn 12288	. . . . 5 ⊢ 3 ∈ ℕ
58		cxproot 26540	. . . . 5 ⊢ ((((𝑁 + (√‘𝐺)) / 2) ∈ ℂ ∧ 3 ∈ ℕ) → ((((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3))↑3) = ((𝑁 + (√‘𝐺)) / 2))
59	50, 57, 58	sylancl 585	. . . 4 ⊢ (𝜑 → ((((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3))↑3) = ((𝑁 + (√‘𝐺)) / 2))
60	56, 59	eqtrd 2764	. . 3 ⊢ (𝜑 → (𝑇↑3) = ((𝑁 + (√‘𝐺)) / 2))
61	47	sqsqrtd 15383	. . . 4 ⊢ (𝜑 → ((√‘𝐺)↑2) = 𝐺)
62	61, 31	eqtrd 2764	. . 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 12317	. . . . . . . . . 10 ⊢ 4 ≠ 0
66	65	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 4 ≠ 0)
67		cubic.0	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≠ 0)
68		3z 12592	. . . . . . . . . . 11 ⊢ 3 ∈ ℤ
69	68	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℤ)
70	41, 67, 69	expne0d 14114	. . . . . . . . 9 ⊢ (𝜑 → (𝑀↑3) ≠ 0)
71	64, 43, 66, 70	mulne0d 11863	. . . . . . . 8 ⊢ (𝜑 → (4 · (𝑀↑3)) ≠ 0)
72	62	oveq2d 7417	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((√‘𝐺)↑2)) = ((𝑁↑2) − ((𝑁↑2) − (4 · (𝑀↑3)))))
73		subsq 14171	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ (√‘𝐺) ∈ ℂ) → ((𝑁↑2) − ((√‘𝐺)↑2)) = ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))))
74	30, 48, 73	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((√‘𝐺)↑2)) = ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))))
75	32, 45	nncand 11573	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((𝑁↑2) − (4 · (𝑀↑3)))) = (4 · (𝑀↑3)))
76	72, 74, 75	3eqtr3d 2772	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) = (4 · (𝑀↑3)))
77	30, 48	subcld 11568	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − (√‘𝐺)) ∈ ℂ)
78	77	mul02d 11409	. . . . . . . 8 ⊢ (𝜑 → (0 · (𝑁 − (√‘𝐺))) = 0)
79	71, 76, 78	3netr4d 3010	. . . . . . 7 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) ≠ (0 · (𝑁 − (√‘𝐺))))
80		oveq1 7408	. . . . . . . 8 ⊢ ((𝑁 + (√‘𝐺)) = 0 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) = (0 · (𝑁 − (√‘𝐺))))
81	80	necon3i 2965	. . . . . . 7 ⊢ (((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) ≠ (0 · (𝑁 − (√‘𝐺))) → (𝑁 + (√‘𝐺)) ≠ 0)
82	79, 81	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 + (√‘𝐺)) ≠ 0)
83		2ne0 12313	. . . . . . 7 ⊢ 2 ≠ 0
84	83	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
85	49, 63, 82, 84	divne0d 12003	. . . . 5 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) / 2) ≠ 0)
86	52	a1i 11	. . . . 5 ⊢ (𝜑 → (1 / 3) ∈ ℂ)
87	50, 85, 86	cxpne0d 26563	. . . 4 ⊢ (𝜑 → (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)) ≠ 0)
88	7, 87	eqnetrd 3000	. . 3 ⊢ (𝜑 → 𝑇 ≠ 0)
89	1, 2, 3, 4, 5, 6, 55, 60, 48, 62, 34, 8, 88	cubic2 26696	. 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 26690	. . . . 5 ⊢ (𝑟 ∈ 𝑅 ↔ (𝑟 ∈ ℂ ∧ (𝑟↑3) = 1))
92	91	anbi1i 623	. . . 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 468	. . . 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 275	. . 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 3082	. 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	bitr4di 289	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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∃wrex 3062 {ctp 4624 ‘cfv 6533 (class class class)co 7401 ℂcc 11104 0cc0 11106 1c1 11107 ici 11108 + caddc 11109 · cmul 11111 − cmin 11441 -cneg 11442 / cdiv 11868 ℕcn 12209 2c2 12264 3c3 12265 4c4 12266 7c7 12269 9c9 12271 ℕ₀cn0 12469 ℤcz 12555 cdc 12674 ↑cexp 14024 √csqrt 15177 ↑_𝑐ccxp 26406

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 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 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-dvds 16195 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-mulg 18986 df-cntz 19223 df-cmn 19692 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-fbas 21225 df-fg 21226 df-cnfld 21229 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-nei 22924 df-lp 22962 df-perf 22963 df-cn 23053 df-cnp 23054 df-haus 23141 df-tx 23388 df-hmeo 23581 df-fil 23672 df-fm 23764 df-flim 23765 df-flf 23766 df-xms 24148 df-ms 24149 df-tms 24150 df-cncf 24720 df-limc 25717 df-dv 25718 df-log 26407 df-cxp 26408

This theorem is referenced by: (None)

W3C validator