requad1 - Mathbox for Alexander van der Vekens

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Theorem requad1 44747

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 10861	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3	2	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℂ)
4		requad2.z	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 0)
5	4	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐴 ≠ 0)
6		requad2.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	recnd 10861	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
8	7	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐵 ∈ ℂ)
9		requad2.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
10	9	recnd 10861	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
11	10	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐶 ∈ ℂ)
12		recn 10819	. . . . . . 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 726	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))
16	3, 5, 8, 11, 13, 15	quad 25723	. . . . 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 3300	. . . 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 11259	. . . . . . . 8 ⊢ (𝜑 → -𝐵 ∈ ℝ)
19	18	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → -𝐵 ∈ ℝ)
20	6	resqcld 13817	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ)
21		4re 11914	. . . . . . . . . . . 12 ⊢ 4 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 4 ∈ ℝ)
23	1, 9	remulcld 10863	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℝ)
24	22, 23	remulcld 10863	. . . . . . . . . 10 ⊢ (𝜑 → (4 · (𝐴 · 𝐶)) ∈ ℝ)
25	20, 24	resubcld 11260	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵↑2) − (4 · (𝐴 · 𝐶))) ∈ ℝ)
26	14, 25	eqeltrd 2838	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
27		resqrtcl 14817	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℝ)
28	26, 27	sylan 583	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℝ)
29	19, 28	readdcld 10862	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-𝐵 + (√‘𝐷)) ∈ ℝ)
30		2re 11904	. . . . . . . . 9 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ)
32	31, 1	remulcld 10863	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ∈ ℝ)
33	32	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ∈ ℝ)
34		2cnd 11908	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
35		2ne0 11934	. . . . . . . . 9 ⊢ 2 ≠ 0
36	35	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ≠ 0)
37	34, 2, 36, 4	mulne0d 11484	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ≠ 0)
38	37	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ≠ 0)
39	29, 33, 38	redivcld 11660	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) ∈ ℝ)
40	19, 28	resubcld 11260	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-𝐵 − (√‘𝐷)) ∈ ℝ)
41	40, 33, 38	redivcld 11660	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) ∈ ℝ)
42		euoreqb 44273	. . . . 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 587	. . . 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 11176	. . . . . . . 8 ⊢ (𝜑 → -𝐵 ∈ ℂ)
45	26	recnd 10861	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℂ)
46	45	sqrtcld 15001	. . . . . . . 8 ⊢ (𝜑 → (√‘𝐷) ∈ ℂ)
47	32	recnd 10861	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐴) ∈ ℂ)
48	44, 46, 47, 37	divdird 11646	. . . . . . 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 11647	. . . . . . . 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 11608	. . . . . . . . 9 ⊢ (𝜑 → (-𝐵 / (2 · 𝐴)) ∈ ℂ)
53	52	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-𝐵 / (2 · 𝐴)) ∈ ℂ)
54	46, 47, 37	divcld 11608	. . . . . . . . 9 ⊢ (𝜑 → ((√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
55	54	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
56	53, 55	negsubd 11195	. . . . . . 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 11621	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → -((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴)))
60	59	oveq2d 7229	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 / (2 · 𝐴)) + -((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))))
61	51, 56, 60	3eqtr2d 2783	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))))
62	49, 61	eqeq12d 2753	. . . . 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 11176	. . . . . . . . 9 ⊢ (𝜑 → -(√‘𝐷) ∈ ℂ)
64	63, 47, 37	divcld 11608	. . . . . . . 8 ⊢ (𝜑 → (-(√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
65	64	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-(√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
66	53, 55, 65	addcand 11035	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))) ↔ ((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴))))
67		div11 11518	. . . . . . . 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 1376	. . . . . . 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 11553	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) = -(√‘𝐷) ↔ (√‘𝐷) = 0))
71		sqrt00 14827	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ ∧ 0 ≤ 𝐷) → ((√‘𝐷) = 0 ↔ 𝐷 = 0))
72	26, 71	sylan 583	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) = 0 ↔ 𝐷 = 0))
73	70, 72	bitrd 282	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) = -(√‘𝐷) ↔ 𝐷 = 0))
74	66, 69, 73	3bitrd 308	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))) ↔ 𝐷 = 0))
75	62, 74	bitrd 282	. . . 4 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) ↔ 𝐷 = 0))
76	17, 43, 75	3bitrd 308	. . 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 44746	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 0 ≤ 𝐷))
79	78	notbid 321	. . . . . . 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 3338	. . . . . 6 ⊢ (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 → ∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0)
82	80, 81	nsyl 142	. . . . 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 10836	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
85	26, 84	ltnled 10979	. . . . . . 7 ⊢ (𝜑 → (𝐷 < 0 ↔ ¬ 0 ≤ 𝐷))
86	85	biimparc 483	. . . . . 6 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → 𝐷 < 0)
87	86	lt0ne0d 11397	. . . . 5 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → 𝐷 ≠ 0)
88		eqneqall 2951	. . . . 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 215	. . 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 185	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 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∃wrex 3062 ∃!wreu 3063 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 + caddc 10732 · cmul 10734 < clt 10867 ≤ cle 10868 − cmin 11062 -cneg 11063 / cdiv 11489 2c2 11885 4c4 11887 ↑cexp 13635 √csqrt 14796

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799

This theorem is referenced by: (None)

W3C validator