requad1 - Mathbox for Alexander van der Vekens

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Theorem requad1 48813

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 11340	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3	2	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℂ)
4		requad2.z	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 0)
5	4	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐴 ≠ 0)
6		requad2.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	recnd 11340	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
8	7	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐵 ∈ ℂ)
9		requad2.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
10	9	recnd 11340	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
11	10	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐶 ∈ ℂ)
12		recn 11293	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
13	12	adantl 487	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
14		requad2.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))
15	14	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))
16	3, 5, 8, 11, 13, 15	quad 27173	. . . . 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 3379	. . . 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 11746	. . . . . . . 8 ⊢ (𝜑 → -𝐵 ∈ ℝ)
19	18	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → -𝐵 ∈ ℝ)
20	6	resqcld 14271	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ)
21		4re 12430	. . . . . . . . . . . 12 ⊢ 4 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 4 ∈ ℝ)
23	1, 9	remulcld 11342	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℝ)
24	22, 23	remulcld 11342	. . . . . . . . . 10 ⊢ (𝜑 → (4 · (𝐴 · 𝐶)) ∈ ℝ)
25	20, 24	resubcld 11747	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵↑2) − (4 · (𝐴 · 𝐶))) ∈ ℝ)
26	14, 25	eqeltrd 2860	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
27		resqrtcl 15425	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℝ)
28	26, 27	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℝ)
29	19, 28	readdcld 11341	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ( -𝐵 + (√‘𝐷)) ∈ ℝ)
30		2re 12420	. . . . . . . . 9 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ)
32	31, 1	remulcld 11342	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ∈ ℝ)
33	32	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ∈ ℝ)
34		2cnd 12424	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
35		2ne0 12452	. . . . . . . . 9 ⊢ 2 ≠ 0
36	35	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ≠ 0)
37	34, 2, 36, 4	mulne0d 11971	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ≠ 0)
38	37	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ≠ 0)
39	29, 33, 38	redivcld 12148	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (( -𝐵 + (√‘𝐷)) / (2 · 𝐴)) ∈ ℝ)
40	19, 28	resubcld 11747	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ( -𝐵 − (√‘𝐷)) ∈ ℝ)
41	40, 33, 38	redivcld 12148	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (( -𝐵 − (√‘𝐷)) / (2 · 𝐴)) ∈ ℝ)
42		euoreqb 48272	. . . . 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 596	. . . 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 11659	. . . . . . . 8 ⊢ (𝜑 → -𝐵 ∈ ℂ)
45	26	recnd 11340	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℂ)
46	45	sqrtcld 15612	. . . . . . . 8 ⊢ (𝜑 → (√‘𝐷) ∈ ℂ)
47	32	recnd 11340	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐴) ∈ ℂ)
48	44, 46, 47, 37	divdird 12134	. . . . . . 7 ⊢ (𝜑 → (( -𝐵 + (√‘𝐷)) / (2 · 𝐴)) = (( -𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))))
49	48	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (( -𝐵 + (√‘𝐷)) / (2 · 𝐴)) = (( -𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))))
50	44, 46, 47, 37	divsubdird 12135	. . . . . . . 8 ⊢ (𝜑 → (( -𝐵 − (√‘𝐷)) / (2 · 𝐴)) = (( -𝐵 / (2 · 𝐴)) − ((√‘𝐷) / (2 · 𝐴))))
51	50	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (( -𝐵 − (√‘𝐷)) / (2 · 𝐴)) = (( -𝐵 / (2 · 𝐴)) − ((√‘𝐷) / (2 · 𝐴))))
52	44, 47, 37	divcld 12096	. . . . . . . . 9 ⊢ (𝜑 → ( -𝐵 / (2 · 𝐴)) ∈ ℂ)
53	52	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ( -𝐵 / (2 · 𝐴)) ∈ ℂ)
54	46, 47, 37	divcld 12096	. . . . . . . . 9 ⊢ (𝜑 → ((√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
55	54	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
56	53, 55	negsubd 11678	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (( -𝐵 / (2 · 𝐴)) + -((√‘𝐷) / (2 · 𝐴))) = (( -𝐵 / (2 · 𝐴)) − ((√‘𝐷) / (2 · 𝐴))))
57	46	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℂ)
58	47	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ∈ ℂ)
59	57, 58, 38	divnegd 12109	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → -((√‘𝐷) / (2 · 𝐴)) = ( -(√‘𝐷) / (2 · 𝐴)))
60	59	oveq2d 7436	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (( -𝐵 / (2 · 𝐴)) + -((√‘𝐷) / (2 · 𝐴))) = (( -𝐵 / (2 · 𝐴)) + ( -(√‘𝐷) / (2 · 𝐴))))
61	51, 56, 60	3eqtr2d 2801	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (( -𝐵 − (√‘𝐷)) / (2 · 𝐴)) = (( -𝐵 / (2 · 𝐴)) + ( -(√‘𝐷) / (2 · 𝐴))))
62	49, 61	eqeq12d 2776	. . . . 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 11659	. . . . . . . . 9 ⊢ (𝜑 → -(√‘𝐷) ∈ ℂ)
64	63, 47, 37	divcld 12096	. . . . . . . 8 ⊢ (𝜑 → ( -(√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
65	64	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ( -(√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
66	53, 55, 65	addcand 11516	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((( -𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))) = (( -𝐵 / (2 · 𝐴)) + ( -(√‘𝐷) / (2 · 𝐴))) ↔ ((√‘𝐷) / (2 · 𝐴)) = ( -(√‘𝐷) / (2 · 𝐴))))
67		div11 12005	. . . . . . . 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 1401	. . . . . . 7 ⊢ (𝜑 → (((√‘𝐷) / (2 · 𝐴)) = ( -(√‘𝐷) / (2 · 𝐴)) ↔ (√‘𝐷) = -(√‘𝐷)))
69	68	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((√‘𝐷) / (2 · 𝐴)) = ( -(√‘𝐷) / (2 · 𝐴)) ↔ (√‘𝐷) = -(√‘𝐷)))
70	57	eqnegd 12041	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) = -(√‘𝐷) ↔ (√‘𝐷) = 0))
71		sqrt00 15435	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ ∧ 0 ≤ 𝐷) → ((√‘𝐷) = 0 ↔ 𝐷 = 0))
72	26, 71	sylan 592	. . . . . . 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 419	. 2 ⊢ (0 ≤ 𝐷 → (𝜑 → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0)))
78	1, 4, 6, 9, 14	requad01 48812	. . . . . . . 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 485	. . . . . 6 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → ¬ ∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0)
81		reurex 3369	. . . . . 6 ⊢ (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 → ∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0)
82	80, 81	nsyl 141	. . . . 5 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → ¬ ∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0)
83	82	pm2.21d 122	. . . 4 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 → 𝐷 = 0))
84		0red 11314	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
85	26, 84	ltnled 11460	. . . . . . 7 ⊢ (𝜑 → (𝐷 < 0 ↔ ¬ 0 ≤ 𝐷))
86	85	biimparc 485	. . . . . 6 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → 𝐷 < 0)
87	86	lt0ne0d 11884	. . . . 5 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → 𝐷 ≠ 0)
88		eqneqall 2966	. . . . 5 ⊢ (𝐷 = 0 → (𝐷 ≠ 0 → ∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0))
89	87, 88	syl5com 32	. . . 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 418	. 2 ⊢ (¬ 0 ≤ 𝐷 → (𝜑 → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0)))
92	77, 91	pm2.61i 184	1 ⊢ (𝜑 → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∃wrex 3086 ∃!wreu 3363 class class class wbr 5102 ‘cfv 6538 (class class class)co 7420 ℂcc 11201 ℝcr 11202 0cc0 11203 + caddc 11206 · cmul 11208 < clt 11346 ≤ cle 11347 − cmin 11544 -cneg 11545 / cdiv 11976 2c2 12400 4c4 12402 ↑cexp 14207 √csqrt 15405

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-er 8720 df-en 8977 df-dom 8978 df-sdom 8979 df-sup 9437 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-n0 12610 df-z 12697 df-uz 12969 df-rp 13124 df-seq 14148 df-exp 14208 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408

This theorem is used by: (None)

W3C validator