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Theorem sqreu 15399

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 15317	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
2	1	recnd 11289	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ)
3		subneg 11558	. . . . . . 7 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((abs‘𝐴) − -𝐴) = ((abs‘𝐴) + 𝐴))
4	2, 3	mpancom 688	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) − -𝐴) = ((abs‘𝐴) + 𝐴))
5	4	eqeq1d 2739	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) − -𝐴) = 0 ↔ ((abs‘𝐴) + 𝐴) = 0))
6		negcl 11508	. . . . . 6 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
7	2, 6	subeq0ad 11630	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) − -𝐴) = 0 ↔ (abs‘𝐴) = -𝐴))
8	5, 7	bitr3d 281	. . . 4 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) = 0 ↔ (abs‘𝐴) = -𝐴))
9		ax-icn 11214	. . . . . . 7 ⊢ i ∈ ℂ
10		absge0 15326	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
11	1, 10	jca 511	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
12		eleq1 2829	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = -𝐴 → ((abs‘𝐴) ∈ ℝ ↔ -𝐴 ∈ ℝ))
13		breq2 5147	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = -𝐴 → (0 ≤ (abs‘𝐴) ↔ 0 ≤ -𝐴))
14	12, 13	anbi12d 632	. . . . . . . . . . 11 ⊢ ((abs‘𝐴) = -𝐴 → (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) ↔ (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴)))
15	11, 14	imbitrid 244	. . . . . . . . . 10 ⊢ ((abs‘𝐴) = -𝐴 → (𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴)))
16	15	impcom 407	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴))
17		resqrtcl 15292	. . . . . . . . 9 ⊢ ((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) → (√‘-𝐴) ∈ ℝ)
18	16, 17	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (√‘-𝐴) ∈ ℝ)
19	18	recnd 11289	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (√‘-𝐴) ∈ ℂ)
20		mulcl 11239	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ (√‘-𝐴) ∈ ℂ) → (i · (√‘-𝐴)) ∈ ℂ)
21	9, 19, 20	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (i · (√‘-𝐴)) ∈ ℂ)
22		sqrtneglem 15305	. . . . . . . 8 ⊢ ((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) → (((i · (√‘-𝐴))↑2) = --𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘-𝐴))) ∧ (i · (i · (√‘-𝐴))) ∉ ℝ⁺))
23	16, 22	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (((i · (√‘-𝐴))↑2) = --𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘-𝐴))) ∧ (i · (i · (√‘-𝐴))) ∉ ℝ⁺))
24		negneg 11559	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴)
25	24	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → --𝐴 = 𝐴)
26	25	eqeq2d 2748	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (((i · (√‘-𝐴))↑2) = --𝐴 ↔ ((i · (√‘-𝐴))↑2) = 𝐴))
27	26	3anbi1d 1442	. . . . . . 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 232	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (((i · (√‘-𝐴))↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘-𝐴))) ∧ (i · (i · (√‘-𝐴))) ∉ ℝ⁺))
29		oveq1 7438	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (𝑥↑2) = ((i · (√‘-𝐴))↑2))
30	29	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑥 = (i · (√‘-𝐴)) → ((𝑥↑2) = 𝐴 ↔ ((i · (√‘-𝐴))↑2) = 𝐴))
31		fveq2 6906	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (ℜ‘𝑥) = (ℜ‘(i · (√‘-𝐴))))
32	31	breq2d 5155	. . . . . . . 8 ⊢ (𝑥 = (i · (√‘-𝐴)) → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘(i · (√‘-𝐴)))))
33		oveq2 7439	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (i · 𝑥) = (i · (i · (√‘-𝐴))))
34		neleq1 3052	. . . . . . . . 9 ⊢ ((i · 𝑥) = (i · (i · (√‘-𝐴))) → ((i · 𝑥) ∉ ℝ⁺ ↔ (i · (i · (√‘-𝐴))) ∉ ℝ⁺))
35	33, 34	syl 17	. . . . . . . 8 ⊢ (𝑥 = (i · (√‘-𝐴)) → ((i · 𝑥) ∉ ℝ⁺ ↔ (i · (i · (√‘-𝐴))) ∉ ℝ⁺))
36	30, 32, 35	3anbi123d 1438	. . . . . . 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 3622	. . . . . 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 584	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
39	38	ex 412	. . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) = -𝐴 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
40	8, 39	sylbid 240	. . 3 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) = 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
41		resqrtcl 15292	. . . . . . . . 9 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) → (√‘(abs‘𝐴)) ∈ ℝ)
42	1, 10, 41	syl2anc 584	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (√‘(abs‘𝐴)) ∈ ℝ)
43	42	recnd 11289	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (√‘(abs‘𝐴)) ∈ ℂ)
44	43	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (√‘(abs‘𝐴)) ∈ ℂ)
45		addcl 11237	. . . . . . . . 9 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((abs‘𝐴) + 𝐴) ∈ ℂ)
46	2, 45	mpancom 688	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + 𝐴) ∈ ℂ)
47	46	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((abs‘𝐴) + 𝐴) ∈ ℂ)
48		abscl 15317	. . . . . . . . . 10 ⊢ (((abs‘𝐴) + 𝐴) ∈ ℂ → (abs‘((abs‘𝐴) + 𝐴)) ∈ ℝ)
49	46, 48	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (abs‘((abs‘𝐴) + 𝐴)) ∈ ℝ)
50	49	recnd 11289	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (abs‘((abs‘𝐴) + 𝐴)) ∈ ℂ)
51	50	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (abs‘((abs‘𝐴) + 𝐴)) ∈ ℂ)
52	46	abs00ad 15329	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((abs‘((abs‘𝐴) + 𝐴)) = 0 ↔ ((abs‘𝐴) + 𝐴) = 0))
53	52	necon3bid 2985	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((abs‘((abs‘𝐴) + 𝐴)) ≠ 0 ↔ ((abs‘𝐴) + 𝐴) ≠ 0))
54	53	biimpar 477	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (abs‘((abs‘𝐴) + 𝐴)) ≠ 0)
55	47, 51, 54	divcld 12043	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))) ∈ ℂ)
56	44, 55	mulcld 11281	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) ∈ ℂ)
57		eqid 2737	. . . . . 6 ⊢ ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))
58	57	sqreulem 15398	. . . . 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 7438	. . . . . . . 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 2739	. . . . . . 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 6906	. . . . . . . 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 5155	. . . . . . 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 7439	. . . . . . . 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 3052	. . . . . . . 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 17	. . . . . . 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 1438	. . . . . 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 3622	. . . . 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 584	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
69	68	ex 412	. . 3 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) ≠ 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
70	40, 69	pm2.61dne 3028	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
71		sqrmo 15290	. 2 ⊢ (𝐴 ∈ ℂ → ∃*𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
72		reu5 3382	. 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 583	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∉ wnel 3046 ∃wrex 3070 ∃!wreu 3378 ∃*wrmo 3379 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 ici 11157 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 -cneg 11493 / cdiv 11920 2c2 12321 ℝ⁺crp 13034 ↑cexp 14102 ℜcre 15136 √csqrt 15272 abscabs 15273

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 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275

This theorem is referenced by: sqrtcl 15400 sqrtthlem 15401 eqsqrtd 15406

W3C validator