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Theorem cubic 26784

Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 4659 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 12197	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
10		3nn0 12396	. . . . . . . . . . . 12 ⊢ 3 ∈ ℕ₀
11		expcl 13983	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐵↑3) ∈ ℂ)
12	3, 10, 11	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵↑3) ∈ ℂ)
13		mulcl 11087	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ (𝐵↑3) ∈ ℂ) → (2 · (𝐵↑3)) ∈ ℂ)
14	9, 12, 13	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (2 · (𝐵↑3)) ∈ ℂ)
15		9cn 12222	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
16		mulcl 11087	. . . . . . . . . . . 12 ⊢ ((9 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (9 · 𝐴) ∈ ℂ)
17	15, 1, 16	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → (9 · 𝐴) ∈ ℂ)
18	3, 4	mulcld 11129	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 · 𝐶) ∈ ℂ)
19	17, 18	mulcld 11129	. . . . . . . . . 10 ⊢ (𝜑 → ((9 · 𝐴) · (𝐵 · 𝐶)) ∈ ℂ)
20	14, 19	subcld 11469	. . . . . . . . 9 ⊢ (𝜑 → ((2 · (𝐵↑3)) − ((9 · 𝐴) · (𝐵 · 𝐶))) ∈ ℂ)
21		2nn0 12395	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
22		7nn 12214	. . . . . . . . . . . 12 ⊢ 7 ∈ ℕ
23	21, 22	decnncl 12605	. . . . . . . . . . 11 ⊢ ;27 ∈ ℕ
24	23	nncni 12132	. . . . . . . . . 10 ⊢ ;27 ∈ ℂ
25	1	sqcld 14048	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ)
26	25, 5	mulcld 11129	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴↑2) · 𝐷) ∈ ℂ)
27		mulcl 11087	. . . . . . . . . 10 ⊢ ((;27 ∈ ℂ ∧ ((𝐴↑2) · 𝐷) ∈ ℂ) → (;27 · ((𝐴↑2) · 𝐷)) ∈ ℂ)
28	24, 26, 27	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (;27 · ((𝐴↑2) · 𝐷)) ∈ ℂ)
29	20, 28	addcld 11128	. . . . . . . 8 ⊢ (𝜑 → (((2 · (𝐵↑3)) − ((9 · 𝐴) · (𝐵 · 𝐶))) + (;27 · ((𝐴↑2) · 𝐷))) ∈ ℂ)
30	8, 29	eqeltrd 2831	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
31		cubic.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((𝑁↑2) − (4 · (𝑀↑3))))
32	30	sqcld 14048	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ)
33		4cn 12207	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
34		cubic.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 = ((𝐵↑2) − (3 · (𝐴 · 𝐶))))
35	3	sqcld 14048	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ)
36		3cn 12203	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ ℂ
37	1, 4	mulcld 11129	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℂ)
38		mulcl 11087	. . . . . . . . . . . . . . 15 ⊢ ((3 ∈ ℂ ∧ (𝐴 · 𝐶) ∈ ℂ) → (3 · (𝐴 · 𝐶)) ∈ ℂ)
39	36, 37, 38	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (3 · (𝐴 · 𝐶)) ∈ ℂ)
40	35, 39	subcld 11469	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵↑2) − (3 · (𝐴 · 𝐶))) ∈ ℂ)
41	34, 40	eqeltrd 2831	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ)
42		expcl 13983	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑀↑3) ∈ ℂ)
43	41, 10, 42	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀↑3) ∈ ℂ)
44		mulcl 11087	. . . . . . . . . . 11 ⊢ ((4 ∈ ℂ ∧ (𝑀↑3) ∈ ℂ) → (4 · (𝑀↑3)) ∈ ℂ)
45	33, 43, 44	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (4 · (𝑀↑3)) ∈ ℂ)
46	32, 45	subcld 11469	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − (4 · (𝑀↑3))) ∈ ℂ)
47	31, 46	eqeltrd 2831	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℂ)
48	47	sqrtcld 15344	. . . . . . 7 ⊢ (𝜑 → (√‘𝐺) ∈ ℂ)
49	30, 48	addcld 11128	. . . . . 6 ⊢ (𝜑 → (𝑁 + (√‘𝐺)) ∈ ℂ)
50	49	halfcld 12363	. . . . 5 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) / 2) ∈ ℂ)
51		3ne0 12228	. . . . . 6 ⊢ 3 ≠ 0
52	36, 51	reccli 11848	. . . . 5 ⊢ (1 / 3) ∈ ℂ
53		cxpcl 26608	. . . . 5 ⊢ ((((𝑁 + (√‘𝐺)) / 2) ∈ ℂ ∧ (1 / 3) ∈ ℂ) → (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)) ∈ ℂ)
54	50, 52, 53	sylancl 586	. . . 4 ⊢ (𝜑 → (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)) ∈ ℂ)
55	7, 54	eqeltrd 2831	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
56	7	oveq1d 7361	. . . 4 ⊢ (𝜑 → (𝑇↑3) = ((((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3))↑3))
57		3nn 12201	. . . . 5 ⊢ 3 ∈ ℕ
58		cxproot 26624	. . . . 5 ⊢ ((((𝑁 + (√‘𝐺)) / 2) ∈ ℂ ∧ 3 ∈ ℕ) → ((((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3))↑3) = ((𝑁 + (√‘𝐺)) / 2))
59	50, 57, 58	sylancl 586	. . . 4 ⊢ (𝜑 → ((((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3))↑3) = ((𝑁 + (√‘𝐺)) / 2))
60	56, 59	eqtrd 2766	. . 3 ⊢ (𝜑 → (𝑇↑3) = ((𝑁 + (√‘𝐺)) / 2))
61	47	sqsqrtd 15346	. . . 4 ⊢ (𝜑 → ((√‘𝐺)↑2) = 𝐺)
62	61, 31	eqtrd 2766	. . 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 12230	. . . . . . . . . 10 ⊢ 4 ≠ 0
66	65	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 4 ≠ 0)
67		cubic.0	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≠ 0)
68		3z 12502	. . . . . . . . . . 11 ⊢ 3 ∈ ℤ
69	68	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℤ)
70	41, 67, 69	expne0d 14056	. . . . . . . . 9 ⊢ (𝜑 → (𝑀↑3) ≠ 0)
71	64, 43, 66, 70	mulne0d 11766	. . . . . . . 8 ⊢ (𝜑 → (4 · (𝑀↑3)) ≠ 0)
72	62	oveq2d 7362	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((√‘𝐺)↑2)) = ((𝑁↑2) − ((𝑁↑2) − (4 · (𝑀↑3)))))
73		subsq 14114	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ (√‘𝐺) ∈ ℂ) → ((𝑁↑2) − ((√‘𝐺)↑2)) = ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))))
74	30, 48, 73	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((√‘𝐺)↑2)) = ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))))
75	32, 45	nncand 11474	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((𝑁↑2) − (4 · (𝑀↑3)))) = (4 · (𝑀↑3)))
76	72, 74, 75	3eqtr3d 2774	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) = (4 · (𝑀↑3)))
77	30, 48	subcld 11469	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − (√‘𝐺)) ∈ ℂ)
78	77	mul02d 11308	. . . . . . . 8 ⊢ (𝜑 → (0 · (𝑁 − (√‘𝐺))) = 0)
79	71, 76, 78	3netr4d 3005	. . . . . . 7 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) ≠ (0 · (𝑁 − (√‘𝐺))))
80		oveq1 7353	. . . . . . . 8 ⊢ ((𝑁 + (√‘𝐺)) = 0 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) = (0 · (𝑁 − (√‘𝐺))))
81	80	necon3i 2960	. . . . . . 7 ⊢ (((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) ≠ (0 · (𝑁 − (√‘𝐺))) → (𝑁 + (√‘𝐺)) ≠ 0)
82	79, 81	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 + (√‘𝐺)) ≠ 0)
83		2ne0 12226	. . . . . . 7 ⊢ 2 ≠ 0
84	83	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
85	49, 63, 82, 84	divne0d 11910	. . . . 5 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) / 2) ≠ 0)
86	52	a1i 11	. . . . 5 ⊢ (𝜑 → (1 / 3) ∈ ℂ)
87	50, 85, 86	cxpne0d 26647	. . . 4 ⊢ (𝜑 → (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)) ≠ 0)
88	7, 87	eqnetrd 2995	. . 3 ⊢ (𝜑 → 𝑇 ≠ 0)
89	1, 2, 3, 4, 5, 6, 55, 60, 48, 62, 34, 8, 88	cubic2 26783	. 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 26777	. . . . 5 ⊢ (𝑟 ∈ 𝑅 ↔ (𝑟 ∈ ℂ ∧ (𝑟↑3) = 1))
92	91	anbi1i 624	. . . 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 3075	. 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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∃wrex 3056 {ctp 4580 ‘cfv 6481 (class class class)co 7346 ℂcc 11001 0cc0 11003 1c1 11004 ici 11005 + caddc 11006 · cmul 11008 − cmin 11341 -cneg 11342 / cdiv 11771 ℕcn 12122 2c2 12177 3c3 12178 4c4 12179 7c7 12182 9c9 12184 ℕ₀cn0 12378 ℤcz 12465 cdc 12585 ↑cexp 13965 √csqrt 15137 ↑_𝑐ccxp 26489

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081 ax-addf 11082

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-q 12844 df-rp 12888 df-xneg 13008 df-xadd 13009 df-xmul 13010 df-ioo 13246 df-ioc 13247 df-ico 13248 df-icc 13249 df-fz 13405 df-fzo 13552 df-fl 13693 df-mod 13771 df-seq 13906 df-exp 13966 df-fac 14178 df-bc 14207 df-hash 14235 df-shft 14971 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-limsup 15375 df-clim 15392 df-rlim 15393 df-sum 15591 df-ef 15971 df-sin 15973 df-cos 15974 df-pi 15976 df-dvds 16161 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-hom 17182 df-cco 17183 df-rest 17323 df-topn 17324 df-0g 17342 df-gsum 17343 df-topgen 17344 df-pt 17345 df-prds 17348 df-xrs 17403 df-qtop 17408 df-imas 17409 df-xps 17411 df-mre 17485 df-mrc 17486 df-acs 17488 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-submnd 18689 df-mulg 18978 df-cntz 19227 df-cmn 19692 df-psmet 21281 df-xmet 21282 df-met 21283 df-bl 21284 df-mopn 21285 df-fbas 21286 df-fg 21287 df-cnfld 21290 df-top 22807 df-topon 22824 df-topsp 22846 df-bases 22859 df-cld 22932 df-ntr 22933 df-cls 22934 df-nei 23011 df-lp 23049 df-perf 23050 df-cn 23140 df-cnp 23141 df-haus 23228 df-tx 23475 df-hmeo 23668 df-fil 23759 df-fm 23851 df-flim 23852 df-flf 23853 df-xms 24233 df-ms 24234 df-tms 24235 df-cncf 24796 df-limc 25792 df-dv 25793 df-log 26490 df-cxp 26491

This theorem is referenced by: (None)

W3C validator