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Theorem cubic 26794

Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 4711 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 12318	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
10		3nn0 12521	. . . . . . . . . . . 12 ⊢ 3 ∈ ℕ₀
11		expcl 14077	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐵↑3) ∈ ℂ)
12	3, 10, 11	sylancl 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵↑3) ∈ ℂ)
13		mulcl 11223	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ (𝐵↑3) ∈ ℂ) → (2 · (𝐵↑3)) ∈ ℂ)
14	9, 12, 13	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → (2 · (𝐵↑3)) ∈ ℂ)
15		9cn 12343	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
16		mulcl 11223	. . . . . . . . . . . 12 ⊢ ((9 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (9 · 𝐴) ∈ ℂ)
17	15, 1, 16	sylancr 586	. . . . . . . . . . 11 ⊢ (𝜑 → (9 · 𝐴) ∈ ℂ)
18	3, 4	mulcld 11265	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 · 𝐶) ∈ ℂ)
19	17, 18	mulcld 11265	. . . . . . . . . 10 ⊢ (𝜑 → ((9 · 𝐴) · (𝐵 · 𝐶)) ∈ ℂ)
20	14, 19	subcld 11602	. . . . . . . . 9 ⊢ (𝜑 → ((2 · (𝐵↑3)) − ((9 · 𝐴) · (𝐵 · 𝐶))) ∈ ℂ)
21		2nn0 12520	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
22		7nn 12335	. . . . . . . . . . . 12 ⊢ 7 ∈ ℕ
23	21, 22	decnncl 12728	. . . . . . . . . . 11 ⊢ ;27 ∈ ℕ
24	23	nncni 12253	. . . . . . . . . 10 ⊢ ;27 ∈ ℂ
25	1	sqcld 14141	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ)
26	25, 5	mulcld 11265	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴↑2) · 𝐷) ∈ ℂ)
27		mulcl 11223	. . . . . . . . . 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 11264	. . . . . . . 8 ⊢ (𝜑 → (((2 · (𝐵↑3)) − ((9 · 𝐴) · (𝐵 · 𝐶))) + (;27 · ((𝐴↑2) · 𝐷))) ∈ ℂ)
30	8, 29	eqeltrd 2829	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
31		cubic.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((𝑁↑2) − (4 · (𝑀↑3))))
32	30	sqcld 14141	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ)
33		4cn 12328	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
34		cubic.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 = ((𝐵↑2) − (3 · (𝐴 · 𝐶))))
35	3	sqcld 14141	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ)
36		3cn 12324	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ ℂ
37	1, 4	mulcld 11265	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℂ)
38		mulcl 11223	. . . . . . . . . . . . . . 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 11602	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵↑2) − (3 · (𝐴 · 𝐶))) ∈ ℂ)
41	34, 40	eqeltrd 2829	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ)
42		expcl 14077	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑀↑3) ∈ ℂ)
43	41, 10, 42	sylancl 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀↑3) ∈ ℂ)
44		mulcl 11223	. . . . . . . . . . 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 11602	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − (4 · (𝑀↑3))) ∈ ℂ)
47	31, 46	eqeltrd 2829	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℂ)
48	47	sqrtcld 15417	. . . . . . 7 ⊢ (𝜑 → (√‘𝐺) ∈ ℂ)
49	30, 48	addcld 11264	. . . . . 6 ⊢ (𝜑 → (𝑁 + (√‘𝐺)) ∈ ℂ)
50	49	halfcld 12488	. . . . 5 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) / 2) ∈ ℂ)
51		3ne0 12349	. . . . . 6 ⊢ 3 ≠ 0
52	36, 51	reccli 11975	. . . . 5 ⊢ (1 / 3) ∈ ℂ
53		cxpcl 26621	. . . . 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 2829	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
56	7	oveq1d 7435	. . . 4 ⊢ (𝜑 → (𝑇↑3) = ((((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3))↑3))
57		3nn 12322	. . . . 5 ⊢ 3 ∈ ℕ
58		cxproot 26637	. . . . 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 2768	. . 3 ⊢ (𝜑 → (𝑇↑3) = ((𝑁 + (√‘𝐺)) / 2))
61	47	sqsqrtd 15419	. . . 4 ⊢ (𝜑 → ((√‘𝐺)↑2) = 𝐺)
62	61, 31	eqtrd 2768	. . 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 12351	. . . . . . . . . 10 ⊢ 4 ≠ 0
66	65	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 4 ≠ 0)
67		cubic.0	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≠ 0)
68		3z 12626	. . . . . . . . . . 11 ⊢ 3 ∈ ℤ
69	68	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℤ)
70	41, 67, 69	expne0d 14149	. . . . . . . . 9 ⊢ (𝜑 → (𝑀↑3) ≠ 0)
71	64, 43, 66, 70	mulne0d 11897	. . . . . . . 8 ⊢ (𝜑 → (4 · (𝑀↑3)) ≠ 0)
72	62	oveq2d 7436	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((√‘𝐺)↑2)) = ((𝑁↑2) − ((𝑁↑2) − (4 · (𝑀↑3)))))
73		subsq 14206	. . . . . . . . . 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 11607	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((𝑁↑2) − (4 · (𝑀↑3)))) = (4 · (𝑀↑3)))
76	72, 74, 75	3eqtr3d 2776	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) = (4 · (𝑀↑3)))
77	30, 48	subcld 11602	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − (√‘𝐺)) ∈ ℂ)
78	77	mul02d 11443	. . . . . . . 8 ⊢ (𝜑 → (0 · (𝑁 − (√‘𝐺))) = 0)
79	71, 76, 78	3netr4d 3015	. . . . . . 7 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) ≠ (0 · (𝑁 − (√‘𝐺))))
80		oveq1 7427	. . . . . . . 8 ⊢ ((𝑁 + (√‘𝐺)) = 0 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) = (0 · (𝑁 − (√‘𝐺))))
81	80	necon3i 2970	. . . . . . 7 ⊢ (((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) ≠ (0 · (𝑁 − (√‘𝐺))) → (𝑁 + (√‘𝐺)) ≠ 0)
82	79, 81	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 + (√‘𝐺)) ≠ 0)
83		2ne0 12347	. . . . . . 7 ⊢ 2 ≠ 0
84	83	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
85	49, 63, 82, 84	divne0d 12037	. . . . 5 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) / 2) ≠ 0)
86	52	a1i 11	. . . . 5 ⊢ (𝜑 → (1 / 3) ∈ ℂ)
87	50, 85, 86	cxpne0d 26660	. . . 4 ⊢ (𝜑 → (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)) ≠ 0)
88	7, 87	eqnetrd 3005	. . 3 ⊢ (𝜑 → 𝑇 ≠ 0)
89	1, 2, 3, 4, 5, 6, 55, 60, 48, 62, 34, 8, 88	cubic2 26793	. 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 26787	. . . . 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 3087	. 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 1534 ∈ wcel 2099 ≠ wne 2937 ∃wrex 3067 {ctp 4633 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 0cc0 11139 1c1 11140 ici 11141 + caddc 11142 · cmul 11144 − cmin 11475 -cneg 11476 / cdiv 11902 ℕcn 12243 2c2 12298 3c3 12299 4c4 12300 7c7 12303 9c9 12305 ℕ₀cn0 12503 ℤcz 12589 cdc 12708 ↑cexp 14059 √csqrt 15213 ↑_𝑐ccxp 26502

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ioc 13362 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-fac 14266 df-bc 14295 df-hash 14323 df-shft 15047 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-ef 16044 df-sin 16046 df-cos 16047 df-pi 16049 df-dvds 16232 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-mulg 19024 df-cntz 19268 df-cmn 19737 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-limc 25808 df-dv 25809 df-log 26503 df-cxp 26504

This theorem is referenced by: (None)

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