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Theorem cubic 26343

Description: The cubic equation, which gives the roots of an arbitrary (nondegenerate) cubic function. Use rextp 4709 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 12283	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
10		3nn0 12486	. . . . . . . . . . . 12 ⊢ 3 ∈ ℕ₀
11		expcl 14041	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐵↑3) ∈ ℂ)
12	3, 10, 11	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵↑3) ∈ ℂ)
13		mulcl 11190	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ (𝐵↑3) ∈ ℂ) → (2 · (𝐵↑3)) ∈ ℂ)
14	9, 12, 13	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (2 · (𝐵↑3)) ∈ ℂ)
15		9cn 12308	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
16		mulcl 11190	. . . . . . . . . . . 12 ⊢ ((9 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (9 · 𝐴) ∈ ℂ)
17	15, 1, 16	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → (9 · 𝐴) ∈ ℂ)
18	3, 4	mulcld 11230	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 · 𝐶) ∈ ℂ)
19	17, 18	mulcld 11230	. . . . . . . . . 10 ⊢ (𝜑 → ((9 · 𝐴) · (𝐵 · 𝐶)) ∈ ℂ)
20	14, 19	subcld 11567	. . . . . . . . 9 ⊢ (𝜑 → ((2 · (𝐵↑3)) − ((9 · 𝐴) · (𝐵 · 𝐶))) ∈ ℂ)
21		2nn0 12485	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
22		7nn 12300	. . . . . . . . . . . 12 ⊢ 7 ∈ ℕ
23	21, 22	decnncl 12693	. . . . . . . . . . 11 ⊢ ;27 ∈ ℕ
24	23	nncni 12218	. . . . . . . . . 10 ⊢ ;27 ∈ ℂ
25	1	sqcld 14105	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ)
26	25, 5	mulcld 11230	. . . . . . . . . 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 587	. . . . . . . . 9 ⊢ (𝜑 → (;27 · ((𝐴↑2) · 𝐷)) ∈ ℂ)
29	20, 28	addcld 11229	. . . . . . . 8 ⊢ (𝜑 → (((2 · (𝐵↑3)) − ((9 · 𝐴) · (𝐵 · 𝐶))) + (;27 · ((𝐴↑2) · 𝐷))) ∈ ℂ)
30	8, 29	eqeltrd 2833	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
31		cubic.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = ((𝑁↑2) − (4 · (𝑀↑3))))
32	30	sqcld 14105	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑2) ∈ ℂ)
33		4cn 12293	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
34		cubic.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 = ((𝐵↑2) − (3 · (𝐴 · 𝐶))))
35	3	sqcld 14105	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ)
36		3cn 12289	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ ℂ
37	1, 4	mulcld 11230	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℂ)
38		mulcl 11190	. . . . . . . . . . . . . . 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 11567	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵↑2) − (3 · (𝐴 · 𝐶))) ∈ ℂ)
41	34, 40	eqeltrd 2833	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ)
42		expcl 14041	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑀↑3) ∈ ℂ)
43	41, 10, 42	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀↑3) ∈ ℂ)
44		mulcl 11190	. . . . . . . . . . 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 11567	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − (4 · (𝑀↑3))) ∈ ℂ)
47	31, 46	eqeltrd 2833	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℂ)
48	47	sqrtcld 15380	. . . . . . 7 ⊢ (𝜑 → (√‘𝐺) ∈ ℂ)
49	30, 48	addcld 11229	. . . . . 6 ⊢ (𝜑 → (𝑁 + (√‘𝐺)) ∈ ℂ)
50	49	halfcld 12453	. . . . 5 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) / 2) ∈ ℂ)
51		3ne0 12314	. . . . . 6 ⊢ 3 ≠ 0
52	36, 51	reccli 11940	. . . . 5 ⊢ (1 / 3) ∈ ℂ
53		cxpcl 26173	. . . . 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 2833	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
56	7	oveq1d 7420	. . . 4 ⊢ (𝜑 → (𝑇↑3) = ((((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3))↑3))
57		3nn 12287	. . . . 5 ⊢ 3 ∈ ℕ
58		cxproot 26189	. . . . 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 2772	. . 3 ⊢ (𝜑 → (𝑇↑3) = ((𝑁 + (√‘𝐺)) / 2))
61	47	sqsqrtd 15382	. . . 4 ⊢ (𝜑 → ((√‘𝐺)↑2) = 𝐺)
62	61, 31	eqtrd 2772	. . 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 12316	. . . . . . . . . 10 ⊢ 4 ≠ 0
66	65	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 4 ≠ 0)
67		cubic.0	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≠ 0)
68		3z 12591	. . . . . . . . . . 11 ⊢ 3 ∈ ℤ
69	68	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℤ)
70	41, 67, 69	expne0d 14113	. . . . . . . . 9 ⊢ (𝜑 → (𝑀↑3) ≠ 0)
71	64, 43, 66, 70	mulne0d 11862	. . . . . . . 8 ⊢ (𝜑 → (4 · (𝑀↑3)) ≠ 0)
72	62	oveq2d 7421	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((√‘𝐺)↑2)) = ((𝑁↑2) − ((𝑁↑2) − (4 · (𝑀↑3)))))
73		subsq 14170	. . . . . . . . . 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 11572	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑2) − ((𝑁↑2) − (4 · (𝑀↑3)))) = (4 · (𝑀↑3)))
76	72, 74, 75	3eqtr3d 2780	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) = (4 · (𝑀↑3)))
77	30, 48	subcld 11567	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − (√‘𝐺)) ∈ ℂ)
78	77	mul02d 11408	. . . . . . . 8 ⊢ (𝜑 → (0 · (𝑁 − (√‘𝐺))) = 0)
79	71, 76, 78	3netr4d 3018	. . . . . . 7 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) ≠ (0 · (𝑁 − (√‘𝐺))))
80		oveq1 7412	. . . . . . . 8 ⊢ ((𝑁 + (√‘𝐺)) = 0 → ((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) = (0 · (𝑁 − (√‘𝐺))))
81	80	necon3i 2973	. . . . . . 7 ⊢ (((𝑁 + (√‘𝐺)) · (𝑁 − (√‘𝐺))) ≠ (0 · (𝑁 − (√‘𝐺))) → (𝑁 + (√‘𝐺)) ≠ 0)
82	79, 81	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 + (√‘𝐺)) ≠ 0)
83		2ne0 12312	. . . . . . 7 ⊢ 2 ≠ 0
84	83	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
85	49, 63, 82, 84	divne0d 12002	. . . . 5 ⊢ (𝜑 → ((𝑁 + (√‘𝐺)) / 2) ≠ 0)
86	52	a1i 11	. . . . 5 ⊢ (𝜑 → (1 / 3) ∈ ℂ)
87	50, 85, 86	cxpne0d 26212	. . . 4 ⊢ (𝜑 → (((𝑁 + (√‘𝐺)) / 2)↑_𝑐(1 / 3)) ≠ 0)
88	7, 87	eqnetrd 3008	. . 3 ⊢ (𝜑 → 𝑇 ≠ 0)
89	1, 2, 3, 4, 5, 6, 55, 60, 48, 62, 34, 8, 88	cubic2 26342	. 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 26336	. . . . 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 469	. . . 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 274	. . 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 3090	. 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 288	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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 {ctp 4631 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 0cc0 11106 1c1 11107 ici 11108 + caddc 11109 · cmul 11111 − cmin 11440 -cneg 11441 / cdiv 11867 ℕcn 12208 2c2 12263 3c3 12264 4c4 12265 7c7 12268 9c9 12270 ℕ₀cn0 12468 ℤcz 12554 cdc 12673 ↑cexp 14023 √csqrt 15176 ↑_𝑐ccxp 26055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 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 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 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 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-dvds 16194 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-cxp 26057

This theorem is referenced by: (None)

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