requad1 - Mathbox for Alexander van der Vekens

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Theorem requad1 48208

Description: A condition for a quadratic equation with real coefficients to have (exactly) one real solution. (Contributed by AV, 26-Jan-2023.)

**Hypotheses**
Ref	Expression
requad2.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
requad2.z	⊢ (𝜑 → 𝐴 ≠ 0)
requad2.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
requad2.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
requad2.d	⊢ (𝜑 → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))

**Assertion**
Ref	Expression
requad1	⊢ (𝜑 → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝑥,𝐷 𝜑,𝑥

**Proof of Theorem requad1**
Step	Hyp	Ref	Expression
1		requad2.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2	1	recnd 11207	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3	2	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℂ)
4		requad2.z	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 0)
5	4	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐴 ≠ 0)
6		requad2.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	recnd 11207	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
8	7	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐵 ∈ ℂ)
9		requad2.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
10	9	recnd 11207	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
11	10	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐶 ∈ ℂ)
12		recn 11160	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
13	12	adantl 485	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
14		requad2.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))
15	14	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))
16	3, 5, 8, 11, 13, 15	quad 26882	. . . . 5 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → (((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ (𝑥 = ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) ∨ 𝑥 = ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)))))
17	16	reubidva 3380	. . . 4 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ ∃!𝑥 ∈ ℝ (𝑥 = ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) ∨ 𝑥 = ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)))))
18	6	renegcld 11611	. . . . . . . 8 ⊢ (𝜑 → -𝐵 ∈ ℝ)
19	18	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → -𝐵 ∈ ℝ)
20	6	resqcld 14135	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ)
21		4re 12299	. . . . . . . . . . . 12 ⊢ 4 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 4 ∈ ℝ)
23	1, 9	remulcld 11209	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℝ)
24	22, 23	remulcld 11209	. . . . . . . . . 10 ⊢ (𝜑 → (4 · (𝐴 · 𝐶)) ∈ ℝ)
25	20, 24	resubcld 11612	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵↑2) − (4 · (𝐴 · 𝐶))) ∈ ℝ)
26	14, 25	eqeltrd 2861	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
27		resqrtcl 15263	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℝ)
28	26, 27	sylan 589	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℝ)
29	19, 28	readdcld 11208	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-𝐵 + (√‘𝐷)) ∈ ℝ)
30		2re 12289	. . . . . . . . 9 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ)
32	31, 1	remulcld 11209	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ∈ ℝ)
33	32	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ∈ ℝ)
34		2cnd 12293	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
35		2ne0 12321	. . . . . . . . 9 ⊢ 2 ≠ 0
36	35	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ≠ 0)
37	34, 2, 36, 4	mulne0d 11836	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ≠ 0)
38	37	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ≠ 0)
39	29, 33, 38	redivcld 12016	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) ∈ ℝ)
40	19, 28	resubcld 11612	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-𝐵 − (√‘𝐷)) ∈ ℝ)
41	40, 33, 38	redivcld 12016	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) ∈ ℝ)
42		euoreqb 47667	. . . . 5 ⊢ ((((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) ∈ ℝ ∧ ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) ∈ ℝ) → (∃!𝑥 ∈ ℝ (𝑥 = ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) ∨ 𝑥 = ((-𝐵 − (√‘𝐷)) / (2 · 𝐴))) ↔ ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 − (√‘𝐷)) / (2 · 𝐴))))
43	39, 41, 42	syl2anc 593	. . . 4 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (∃!𝑥 ∈ ℝ (𝑥 = ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) ∨ 𝑥 = ((-𝐵 − (√‘𝐷)) / (2 · 𝐴))) ↔ ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 − (√‘𝐷)) / (2 · 𝐴))))
44	7	negcld 11526	. . . . . . . 8 ⊢ (𝜑 → -𝐵 ∈ ℂ)
45	26	recnd 11207	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℂ)
46	45	sqrtcld 15450	. . . . . . . 8 ⊢ (𝜑 → (√‘𝐷) ∈ ℂ)
47	32	recnd 11207	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐴) ∈ ℂ)
48	44, 46, 47, 37	divdird 12002	. . . . . . 7 ⊢ (𝜑 → ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))))
49	48	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))))
50	44, 46, 47, 37	divsubdird 12003	. . . . . . . 8 ⊢ (𝜑 → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) − ((√‘𝐷) / (2 · 𝐴))))
51	50	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) − ((√‘𝐷) / (2 · 𝐴))))
52	44, 47, 37	divcld 11964	. . . . . . . . 9 ⊢ (𝜑 → (-𝐵 / (2 · 𝐴)) ∈ ℂ)
53	52	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-𝐵 / (2 · 𝐴)) ∈ ℂ)
54	46, 47, 37	divcld 11964	. . . . . . . . 9 ⊢ (𝜑 → ((√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
55	54	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
56	53, 55	negsubd 11545	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 / (2 · 𝐴)) + -((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) − ((√‘𝐷) / (2 · 𝐴))))
57	46	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℂ)
58	47	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ∈ ℂ)
59	57, 58, 38	divnegd 11977	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → -((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴)))
60	59	oveq2d 7408	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 / (2 · 𝐴)) + -((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))))
61	51, 56, 60	3eqtr2d 2802	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))))
62	49, 61	eqeq12d 2777	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) ↔ ((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴)))))
63	46	negcld 11526	. . . . . . . . 9 ⊢ (𝜑 → -(√‘𝐷) ∈ ℂ)
64	63, 47, 37	divcld 11964	. . . . . . . 8 ⊢ (𝜑 → (-(√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
65	64	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-(√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
66	53, 55, 65	addcand 11383	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))) ↔ ((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴))))
67		div11 11870	. . . . . . . 8 ⊢ (((√‘𝐷) ∈ ℂ ∧ -(√‘𝐷) ∈ ℂ ∧ ((2 · 𝐴) ∈ ℂ ∧ (2 · 𝐴) ≠ 0)) → (((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴)) ↔ (√‘𝐷) = -(√‘𝐷)))
68	46, 63, 47, 37, 67	syl112anc 1392	. . . . . . 7 ⊢ (𝜑 → (((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴)) ↔ (√‘𝐷) = -(√‘𝐷)))
69	68	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴)) ↔ (√‘𝐷) = -(√‘𝐷)))
70	57	eqnegd 11909	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) = -(√‘𝐷) ↔ (√‘𝐷) = 0))
71		sqrt00 15273	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ ∧ 0 ≤ 𝐷) → ((√‘𝐷) = 0 ↔ 𝐷 = 0))
72	26, 71	sylan 589	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) = 0 ↔ 𝐷 = 0))
73	70, 72	bitrd 281	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) = -(√‘𝐷) ↔ 𝐷 = 0))
74	66, 69, 73	3bitrd 307	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))) ↔ 𝐷 = 0))
75	62, 74	bitrd 281	. . . 4 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) ↔ 𝐷 = 0))
76	17, 43, 75	3bitrd 307	. . 3 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0))
77	76	expcom 417	. 2 ⊢ (0 ≤ 𝐷 → (𝜑 → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0)))
78	1, 4, 6, 9, 14	requad01 48207	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 0 ≤ 𝐷))
79	78	notbid 320	. . . . . . 7 ⊢ (𝜑 → (¬ ∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ ¬ 0 ≤ 𝐷))
80	79	biimparc 483	. . . . . 6 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → ¬ ∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0)
81		reurex 3370	. . . . . 6 ⊢ (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 → ∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0)
82	80, 81	nsyl 140	. . . . 5 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → ¬ ∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0)
83	82	pm2.21d 121	. . . 4 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 → 𝐷 = 0))
84		0red 11181	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
85	26, 84	ltnled 11327	. . . . . . 7 ⊢ (𝜑 → (𝐷 < 0 ↔ ¬ 0 ≤ 𝐷))
86	85	biimparc 483	. . . . . 6 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → 𝐷 < 0)
87	86	lt0ne0d 11749	. . . . 5 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → 𝐷 ≠ 0)
88		eqneqall 2967	. . . . 5 ⊢ (𝐷 = 0 → (𝐷 ≠ 0 → ∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0))
89	87, 88	syl5com 31	. . . 4 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → (𝐷 = 0 → ∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0))
90	83, 89	impbid 214	. . 3 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0))
91	90	ex 416	. 2 ⊢ (¬ 0 ≤ 𝐷 → (𝜑 → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0)))
92	77, 91	pm2.61i 183	1 ⊢ (𝜑 → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∃wrex 3085 ∃!wreu 3364 class class class wbr 5099 ‘cfv 6517 (class class class)co 7392 ℂcc 11068 ℝcr 11069 0cc0 11070 + caddc 11073 · cmul 11075 < clt 11213 ≤ cle 11214 − cmin 11411 -cneg 11412 / cdiv 11841 2c2 12269 4c4 12271 ↑cexp 14071 √csqrt 15243

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-sup 9385 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-n0 12479 df-z 12566 df-uz 12837 df-rp 12991 df-seq 14012 df-exp 14072 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246

This theorem is referenced by: (None)

W3C validator