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Theorem sqreu 15408

Description: Existence and uniqueness for the square root function in general. (Contributed by Mario Carneiro, 9-Jul-2013.)

**Assertion**
Ref	Expression
sqreu	⊢ (𝐴 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))

Distinct variable group: 𝑥,𝐴

**Proof of Theorem sqreu**
Step	Hyp	Ref	Expression
1		abscl 15325	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
2	1	recnd 11232	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ)
3		subneg 11502	. . . . . . 7 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((abs‘𝐴) − -𝐴) = ((abs‘𝐴) + 𝐴))
4	2, 3	mpancom 700	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) − -𝐴) = ((abs‘𝐴) + 𝐴))
5	4	eqeq1d 2765	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) − -𝐴) = 0 ↔ ((abs‘𝐴) + 𝐴) = 0))
6		negcl 11452	. . . . . 6 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
7	2, 6	subeq0ad 11574	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) − -𝐴) = 0 ↔ (abs‘𝐴) = -𝐴))
8	5, 7	bitr3d 284	. . . 4 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) = 0 ↔ (abs‘𝐴) = -𝐴))
9		ax-icn 11154	. . . . . . 7 ⊢ i ∈ ℂ
10		absge0 15334	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
11	1, 10	jca 520	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
12		eleq1 2851	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = -𝐴 → ((abs‘𝐴) ∈ ℝ ↔ -𝐴 ∈ ℝ))
13		breq2 5113	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = -𝐴 → (0 ≤ (abs‘𝐴) ↔ 0 ≤ -𝐴))
14	12, 13	anbi12d 643	. . . . . . . . . . 11 ⊢ ((abs‘𝐴) = -𝐴 → (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) ↔ (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴)))
15	11, 14	imbitrid 247	. . . . . . . . . 10 ⊢ ((abs‘𝐴) = -𝐴 → (𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴)))
16	15	impcom 412	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴))
17		resqrtcl 15300	. . . . . . . . 9 ⊢ ((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) → (√‘-𝐴) ∈ ℝ)
18	16, 17	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (√‘-𝐴) ∈ ℝ)
19	18	recnd 11232	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (√‘-𝐴) ∈ ℂ)
20		mulcl 11179	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ (√‘-𝐴) ∈ ℂ) → (i · (√‘-𝐴)) ∈ ℂ)
21	9, 19, 20	sylancr 598	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (i · (√‘-𝐴)) ∈ ℂ)
22		sqrtneglem 15313	. . . . . . . 8 ⊢ ((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) → (((i · (√‘-𝐴))↑2) = --𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘-𝐴))) ∧ (i · (i · (√‘-𝐴))) ∉ ℝ⁺))
23	16, 22	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (((i · (√‘-𝐴))↑2) = --𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘-𝐴))) ∧ (i · (i · (√‘-𝐴))) ∉ ℝ⁺))
24		negneg 11503	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴)
25	24	adantr 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → --𝐴 = 𝐴)
26	25	eqeq2d 2774	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (((i · (√‘-𝐴))↑2) = --𝐴 ↔ ((i · (√‘-𝐴))↑2) = 𝐴))
27	26	3anbi1d 1468	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → ((((i · (√‘-𝐴))↑2) = --𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘-𝐴))) ∧ (i · (i · (√‘-𝐴))) ∉ ℝ⁺) ↔ (((i · (√‘-𝐴))↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘-𝐴))) ∧ (i · (i · (√‘-𝐴))) ∉ ℝ⁺)))
28	23, 27	mpbid 235	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (((i · (√‘-𝐴))↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘-𝐴))) ∧ (i · (i · (√‘-𝐴))) ∉ ℝ⁺))
29		oveq1 7417	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (𝑥↑2) = ((i · (√‘-𝐴))↑2))
30	29	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑥 = (i · (√‘-𝐴)) → ((𝑥↑2) = 𝐴 ↔ ((i · (√‘-𝐴))↑2) = 𝐴))
31		fveq2 6881	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (ℜ‘𝑥) = (ℜ‘(i · (√‘-𝐴))))
32	31	breq2d 5121	. . . . . . . 8 ⊢ (𝑥 = (i · (√‘-𝐴)) → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘(i · (√‘-𝐴)))))
33		oveq2 7418	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (i · 𝑥) = (i · (i · (√‘-𝐴))))
34		neleq1 3070	. . . . . . . . 9 ⊢ ((i · 𝑥) = (i · (i · (√‘-𝐴))) → ((i · 𝑥) ∉ ℝ⁺ ↔ (i · (i · (√‘-𝐴))) ∉ ℝ⁺))
35	33, 34	syl 18	. . . . . . . 8 ⊢ (𝑥 = (i · (√‘-𝐴)) → ((i · 𝑥) ∉ ℝ⁺ ↔ (i · (i · (√‘-𝐴))) ∉ ℝ⁺))
36	30, 32, 35	3anbi123d 1464	. . . . . . 7 ⊢ (𝑥 = (i · (√‘-𝐴)) → (((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺) ↔ (((i · (√‘-𝐴))↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘-𝐴))) ∧ (i · (i · (√‘-𝐴))) ∉ ℝ⁺)))
37	36	rspcev 3581	. . . . . 6 ⊢ (((i · (√‘-𝐴)) ∈ ℂ ∧ (((i · (√‘-𝐴))↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘-𝐴))) ∧ (i · (i · (√‘-𝐴))) ∉ ℝ⁺)) → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
38	21, 28, 37	syl2anc 595	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
39	38	ex 417	. . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) = -𝐴 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
40	8, 39	sylbid 243	. . 3 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) = 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
41		resqrtcl 15300	. . . . . . . . 9 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) → (√‘(abs‘𝐴)) ∈ ℝ)
42	1, 10, 41	syl2anc 595	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (√‘(abs‘𝐴)) ∈ ℝ)
43	42	recnd 11232	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (√‘(abs‘𝐴)) ∈ ℂ)
44	43	adantr 485	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (√‘(abs‘𝐴)) ∈ ℂ)
45		addcl 11177	. . . . . . . . 9 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((abs‘𝐴) + 𝐴) ∈ ℂ)
46	2, 45	mpancom 700	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + 𝐴) ∈ ℂ)
47	46	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((abs‘𝐴) + 𝐴) ∈ ℂ)
48		abscl 15325	. . . . . . . . . 10 ⊢ (((abs‘𝐴) + 𝐴) ∈ ℂ → (abs‘((abs‘𝐴) + 𝐴)) ∈ ℝ)
49	46, 48	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (abs‘((abs‘𝐴) + 𝐴)) ∈ ℝ)
50	49	recnd 11232	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (abs‘((abs‘𝐴) + 𝐴)) ∈ ℂ)
51	50	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (abs‘((abs‘𝐴) + 𝐴)) ∈ ℂ)
52	46	abs00ad 15337	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((abs‘((abs‘𝐴) + 𝐴)) = 0 ↔ ((abs‘𝐴) + 𝐴) = 0))
53	52	necon3bid 3002	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((abs‘((abs‘𝐴) + 𝐴)) ≠ 0 ↔ ((abs‘𝐴) + 𝐴) ≠ 0))
54	53	biimpar 482	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (abs‘((abs‘𝐴) + 𝐴)) ≠ 0)
55	47, 51, 54	divcld 11986	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))) ∈ ℂ)
56	44, 55	mulcld 11224	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) ∈ ℂ)
57		eqid 2763	. . . . . 6 ⊢ ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))
58	57	sqreulem 15407	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))↑2) = 𝐴 ∧ 0 ≤ (ℜ‘((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))) ∧ (i · ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))) ∉ ℝ⁺))
59		oveq1 7417	. . . . . . . 8 ⊢ (𝑥 = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) → (𝑥↑2) = (((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))↑2))
60	59	eqeq1d 2765	. . . . . . 7 ⊢ (𝑥 = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) → ((𝑥↑2) = 𝐴 ↔ (((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))↑2) = 𝐴))
61		fveq2 6881	. . . . . . . 8 ⊢ (𝑥 = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) → (ℜ‘𝑥) = (ℜ‘((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))))
62	61	breq2d 5121	. . . . . . 7 ⊢ (𝑥 = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))))))
63		oveq2 7418	. . . . . . . 8 ⊢ (𝑥 = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) → (i · 𝑥) = (i · ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))))
64		neleq1 3070	. . . . . . . 8 ⊢ ((i · 𝑥) = (i · ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))) → ((i · 𝑥) ∉ ℝ⁺ ↔ (i · ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))) ∉ ℝ⁺))
65	63, 64	syl 18	. . . . . . 7 ⊢ (𝑥 = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) → ((i · 𝑥) ∉ ℝ⁺ ↔ (i · ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))) ∉ ℝ⁺))
66	60, 62, 65	3anbi123d 1464	. . . . . 6 ⊢ (𝑥 = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) → (((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺) ↔ ((((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))↑2) = 𝐴 ∧ 0 ≤ (ℜ‘((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))) ∧ (i · ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))) ∉ ℝ⁺)))
67	66	rspcev 3581	. . . . 5 ⊢ ((((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) ∈ ℂ ∧ ((((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))↑2) = 𝐴 ∧ 0 ≤ (ℜ‘((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))) ∧ (i · ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))) ∉ ℝ⁺)) → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
68	56, 58, 67	syl2anc 595	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
69	68	ex 417	. . 3 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) ≠ 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
70	40, 69	pm2.61dne 3044	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
71		sqrmo 15298	. 2 ⊢ (𝐴 ∈ ℂ → ∃*𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
72		reu5 3371	. 2 ⊢ (∃!𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺) ↔ (∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺) ∧ ∃*𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
73	70, 71, 72	sylanbrc 594	1 ⊢ (𝐴 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∉ wnel 3064 ∃wrex 3089 ∃!wreu 3367 ∃*wrmo 3368 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 0cc0 11095 ici 11097 + caddc 11098 · cmul 11100 ≤ cle 11239 − cmin 11436 -cneg 11437 / cdiv 11866 2c2 12290 ℝ⁺crp 13011 ↑cexp 14093 ℜcre 15144 √csqrt 15280 abscabs 15281

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9398 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-n0 12500 df-z 12587 df-uz 12858 df-rp 13012 df-seq 14034 df-exp 14094 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283

This theorem is referenced by: sqrtcl 15409 sqrtthlem 15410 eqsqrtd 15415

W3C validator