requad1 - Mathbox for Alexander van der Vekens

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Theorem requad1 45074

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 11003	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3	2	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℂ)
4		requad2.z	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 0)
5	4	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐴 ≠ 0)
6		requad2.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	recnd 11003	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
8	7	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐵 ∈ ℂ)
9		requad2.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
10	9	recnd 11003	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
11	10	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐶 ∈ ℂ)
12		recn 10961	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
13	12	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
14		requad2.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))
15	14	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))
16	3, 5, 8, 11, 13, 15	quad 25990	. . . . 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 3322	. . . 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 11402	. . . . . . . 8 ⊢ (𝜑 → -𝐵 ∈ ℝ)
19	18	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → -𝐵 ∈ ℝ)
20	6	resqcld 13965	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ)
21		4re 12057	. . . . . . . . . . . 12 ⊢ 4 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 4 ∈ ℝ)
23	1, 9	remulcld 11005	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℝ)
24	22, 23	remulcld 11005	. . . . . . . . . 10 ⊢ (𝜑 → (4 · (𝐴 · 𝐶)) ∈ ℝ)
25	20, 24	resubcld 11403	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵↑2) − (4 · (𝐴 · 𝐶))) ∈ ℝ)
26	14, 25	eqeltrd 2839	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
27		resqrtcl 14965	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℝ)
28	26, 27	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℝ)
29	19, 28	readdcld 11004	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-𝐵 + (√‘𝐷)) ∈ ℝ)
30		2re 12047	. . . . . . . . 9 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ)
32	31, 1	remulcld 11005	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ∈ ℝ)
33	32	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ∈ ℝ)
34		2cnd 12051	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
35		2ne0 12077	. . . . . . . . 9 ⊢ 2 ≠ 0
36	35	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ≠ 0)
37	34, 2, 36, 4	mulne0d 11627	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ≠ 0)
38	37	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ≠ 0)
39	29, 33, 38	redivcld 11803	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) ∈ ℝ)
40	19, 28	resubcld 11403	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-𝐵 − (√‘𝐷)) ∈ ℝ)
41	40, 33, 38	redivcld 11803	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) ∈ ℝ)
42		euoreqb 44601	. . . . 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 584	. . . 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 11319	. . . . . . . 8 ⊢ (𝜑 → -𝐵 ∈ ℂ)
45	26	recnd 11003	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℂ)
46	45	sqrtcld 15149	. . . . . . . 8 ⊢ (𝜑 → (√‘𝐷) ∈ ℂ)
47	32	recnd 11003	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐴) ∈ ℂ)
48	44, 46, 47, 37	divdird 11789	. . . . . . 7 ⊢ (𝜑 → ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))))
49	48	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))))
50	44, 46, 47, 37	divsubdird 11790	. . . . . . . 8 ⊢ (𝜑 → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) − ((√‘𝐷) / (2 · 𝐴))))
51	50	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) − ((√‘𝐷) / (2 · 𝐴))))
52	44, 47, 37	divcld 11751	. . . . . . . . 9 ⊢ (𝜑 → (-𝐵 / (2 · 𝐴)) ∈ ℂ)
53	52	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-𝐵 / (2 · 𝐴)) ∈ ℂ)
54	46, 47, 37	divcld 11751	. . . . . . . . 9 ⊢ (𝜑 → ((√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
55	54	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
56	53, 55	negsubd 11338	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 / (2 · 𝐴)) + -((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) − ((√‘𝐷) / (2 · 𝐴))))
57	46	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℂ)
58	47	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ∈ ℂ)
59	57, 58, 38	divnegd 11764	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → -((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴)))
60	59	oveq2d 7291	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 / (2 · 𝐴)) + -((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))))
61	51, 56, 60	3eqtr2d 2784	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))))
62	49, 61	eqeq12d 2754	. . . . 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 11319	. . . . . . . . 9 ⊢ (𝜑 → -(√‘𝐷) ∈ ℂ)
64	63, 47, 37	divcld 11751	. . . . . . . 8 ⊢ (𝜑 → (-(√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
65	64	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-(√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
66	53, 55, 65	addcand 11178	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))) ↔ ((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴))))
67		div11 11661	. . . . . . . 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 1373	. . . . . . 7 ⊢ (𝜑 → (((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴)) ↔ (√‘𝐷) = -(√‘𝐷)))
69	68	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴)) ↔ (√‘𝐷) = -(√‘𝐷)))
70	57	eqnegd 11696	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) = -(√‘𝐷) ↔ (√‘𝐷) = 0))
71		sqrt00 14975	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ ∧ 0 ≤ 𝐷) → ((√‘𝐷) = 0 ↔ 𝐷 = 0))
72	26, 71	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) = 0 ↔ 𝐷 = 0))
73	70, 72	bitrd 278	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) = -(√‘𝐷) ↔ 𝐷 = 0))
74	66, 69, 73	3bitrd 305	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))) ↔ 𝐷 = 0))
75	62, 74	bitrd 278	. . . 4 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) ↔ 𝐷 = 0))
76	17, 43, 75	3bitrd 305	. . 3 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0))
77	76	expcom 414	. 2 ⊢ (0 ≤ 𝐷 → (𝜑 → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0)))
78	1, 4, 6, 9, 14	requad01 45073	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 0 ≤ 𝐷))
79	78	notbid 318	. . . . . . 7 ⊢ (𝜑 → (¬ ∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ ¬ 0 ≤ 𝐷))
80	79	biimparc 480	. . . . . 6 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → ¬ ∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0)
81		reurex 3362	. . . . . 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 10978	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
85	26, 84	ltnled 11122	. . . . . . 7 ⊢ (𝜑 → (𝐷 < 0 ↔ ¬ 0 ≤ 𝐷))
86	85	biimparc 480	. . . . . 6 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → 𝐷 < 0)
87	86	lt0ne0d 11540	. . . . 5 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → 𝐷 ≠ 0)
88		eqneqall 2954	. . . . 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 211	. . 3 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0))
91	90	ex 413	. 2 ⊢ (¬ 0 ≤ 𝐷 → (𝜑 → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0)))
92	77, 91	pm2.61i 182	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 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 ∃!wreu 3066 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 2c2 12028 4c4 12030 ↑cexp 13782 √csqrt 14944

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947

This theorem is referenced by: (None)

W3C validator