requad1 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > requad1		Structured version Visualization version GIF version

Theorem requad1 47636

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 11263	. . . . . . 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 11263	. . . . . . 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 11263	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
11	10	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 0 ≤ 𝐷) ∧ 𝑥 ∈ ℝ) → 𝐶 ∈ ℂ)
12		recn 11219	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
13	12	adantl 481	. . . . . 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 26802	. . . . 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 3375	. . . 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 11664	. . . . . . . 8 ⊢ (𝜑 → -𝐵 ∈ ℝ)
19	18	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → -𝐵 ∈ ℝ)
20	6	resqcld 14143	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ)
21		4re 12324	. . . . . . . . . . . 12 ⊢ 4 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 4 ∈ ℝ)
23	1, 9	remulcld 11265	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℝ)
24	22, 23	remulcld 11265	. . . . . . . . . 10 ⊢ (𝜑 → (4 · (𝐴 · 𝐶)) ∈ ℝ)
25	20, 24	resubcld 11665	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵↑2) − (4 · (𝐴 · 𝐶))) ∈ ℝ)
26	14, 25	eqeltrd 2834	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ)
27		resqrtcl 15272	. . . . . . . 8 ⊢ ((𝐷 ∈ ℝ ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℝ)
28	26, 27	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℝ)
29	19, 28	readdcld 11264	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-𝐵 + (√‘𝐷)) ∈ ℝ)
30		2re 12314	. . . . . . . . 9 ⊢ 2 ∈ ℝ
31	30	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ)
32	31, 1	remulcld 11265	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ∈ ℝ)
33	32	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ∈ ℝ)
34		2cnd 12318	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
35		2ne0 12344	. . . . . . . . 9 ⊢ 2 ≠ 0
36	35	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ≠ 0)
37	34, 2, 36, 4	mulne0d 11889	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐴) ≠ 0)
38	37	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ≠ 0)
39	29, 33, 38	redivcld 12069	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) ∈ ℝ)
40	19, 28	resubcld 11665	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-𝐵 − (√‘𝐷)) ∈ ℝ)
41	40, 33, 38	redivcld 12069	. . . . 5 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) ∈ ℝ)
42		euoreqb 47138	. . . . 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 11581	. . . . . . . 8 ⊢ (𝜑 → -𝐵 ∈ ℂ)
45	26	recnd 11263	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℂ)
46	45	sqrtcld 15456	. . . . . . . 8 ⊢ (𝜑 → (√‘𝐷) ∈ ℂ)
47	32	recnd 11263	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐴) ∈ ℂ)
48	44, 46, 47, 37	divdird 12055	. . . . . . 7 ⊢ (𝜑 → ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))))
49	48	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 + (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))))
50	44, 46, 47, 37	divsubdird 12056	. . . . . . . 8 ⊢ (𝜑 → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) − ((√‘𝐷) / (2 · 𝐴))))
51	50	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) − ((√‘𝐷) / (2 · 𝐴))))
52	44, 47, 37	divcld 12017	. . . . . . . . 9 ⊢ (𝜑 → (-𝐵 / (2 · 𝐴)) ∈ ℂ)
53	52	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-𝐵 / (2 · 𝐴)) ∈ ℂ)
54	46, 47, 37	divcld 12017	. . . . . . . . 9 ⊢ (𝜑 → ((√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
55	54	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
56	53, 55	negsubd 11600	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 / (2 · 𝐴)) + -((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) − ((√‘𝐷) / (2 · 𝐴))))
57	46	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (√‘𝐷) ∈ ℂ)
58	47	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (2 · 𝐴) ∈ ℂ)
59	57, 58, 38	divnegd 12030	. . . . . . . 8 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → -((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴)))
60	59	oveq2d 7421	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 / (2 · 𝐴)) + -((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))))
61	51, 56, 60	3eqtr2d 2776	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((-𝐵 − (√‘𝐷)) / (2 · 𝐴)) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))))
62	49, 61	eqeq12d 2751	. . . . 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 11581	. . . . . . . . 9 ⊢ (𝜑 → -(√‘𝐷) ∈ ℂ)
64	63, 47, 37	divcld 12017	. . . . . . . 8 ⊢ (𝜑 → (-(√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
65	64	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (-(√‘𝐷) / (2 · 𝐴)) ∈ ℂ)
66	53, 55, 65	addcand 11438	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((-𝐵 / (2 · 𝐴)) + ((√‘𝐷) / (2 · 𝐴))) = ((-𝐵 / (2 · 𝐴)) + (-(√‘𝐷) / (2 · 𝐴))) ↔ ((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴))))
67		div11 11924	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → (((√‘𝐷) / (2 · 𝐴)) = (-(√‘𝐷) / (2 · 𝐴)) ↔ (√‘𝐷) = -(√‘𝐷)))
70	57	eqnegd 11962	. . . . . . 7 ⊢ ((𝜑 ∧ 0 ≤ 𝐷) → ((√‘𝐷) = -(√‘𝐷) ↔ (√‘𝐷) = 0))
71		sqrt00 15282	. . . . . . . 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 279	. . . . . 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 279	. . . 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 413	. 2 ⊢ (0 ≤ 𝐷 → (𝜑 → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0)))
78	1, 4, 6, 9, 14	requad01 47635	. . . . . . . 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 479	. . . . . 6 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → ¬ ∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0)
81		reurex 3363	. . . . . 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 11238	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
85	26, 84	ltnled 11382	. . . . . . 7 ⊢ (𝜑 → (𝐷 < 0 ↔ ¬ 0 ≤ 𝐷))
86	85	biimparc 479	. . . . . 6 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → 𝐷 < 0)
87	86	lt0ne0d 11802	. . . . 5 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → 𝐷 ≠ 0)
88		eqneqall 2943	. . . . 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 212	. . 3 ⊢ ((¬ 0 ≤ 𝐷 ∧ 𝜑) → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0))
91	90	ex 412	. 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 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∃wrex 3060 ∃!wreu 3357 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ℝcr 11128 0cc0 11129 + caddc 11132 · cmul 11134 < clt 11269 ≤ cle 11270 − cmin 11466 -cneg 11467 / cdiv 11894 2c2 12295 4c4 12297 ↑cexp 14079 √csqrt 15252

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-sup 9454 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-n0 12502 df-z 12589 df-uz 12853 df-rp 13009 df-seq 14020 df-exp 14080 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255

This theorem is referenced by: (None)

W3C validator