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Theorem sqreu 15072

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 14990	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
2	1	recnd 11003	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ)
3		subneg 11270	. . . . . . 7 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((abs‘𝐴) − -𝐴) = ((abs‘𝐴) + 𝐴))
4	2, 3	mpancom 685	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) − -𝐴) = ((abs‘𝐴) + 𝐴))
5	4	eqeq1d 2740	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) − -𝐴) = 0 ↔ ((abs‘𝐴) + 𝐴) = 0))
6		negcl 11221	. . . . . 6 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
7	2, 6	subeq0ad 11342	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) − -𝐴) = 0 ↔ (abs‘𝐴) = -𝐴))
8	5, 7	bitr3d 280	. . . 4 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) = 0 ↔ (abs‘𝐴) = -𝐴))
9		ax-icn 10930	. . . . . . 7 ⊢ i ∈ ℂ
10		absge0 14999	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
11	1, 10	jca 512	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
12		eleq1 2826	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = -𝐴 → ((abs‘𝐴) ∈ ℝ ↔ -𝐴 ∈ ℝ))
13		breq2 5078	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = -𝐴 → (0 ≤ (abs‘𝐴) ↔ 0 ≤ -𝐴))
14	12, 13	anbi12d 631	. . . . . . . . . . 11 ⊢ ((abs‘𝐴) = -𝐴 → (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) ↔ (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴)))
15	11, 14	syl5ib 243	. . . . . . . . . 10 ⊢ ((abs‘𝐴) = -𝐴 → (𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴)))
16	15	impcom 408	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴))
17		resqrtcl 14965	. . . . . . . . 9 ⊢ ((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) → (√‘-𝐴) ∈ ℝ)
18	16, 17	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (√‘-𝐴) ∈ ℝ)
19	18	recnd 11003	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (√‘-𝐴) ∈ ℂ)
20		mulcl 10955	. . . . . . 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 14978	. . . . . . . 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 11271	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴)
25	24	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → --𝐴 = 𝐴)
26	25	eqeq2d 2749	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (((i · (√‘-𝐴))↑2) = --𝐴 ↔ ((i · (√‘-𝐴))↑2) = 𝐴))
27	26	3anbi1d 1439	. . . . . . 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 231	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (((i · (√‘-𝐴))↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘-𝐴))) ∧ (i · (i · (√‘-𝐴))) ∉ ℝ⁺))
29		oveq1 7282	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (𝑥↑2) = ((i · (√‘-𝐴))↑2))
30	29	eqeq1d 2740	. . . . . . . 8 ⊢ (𝑥 = (i · (√‘-𝐴)) → ((𝑥↑2) = 𝐴 ↔ ((i · (√‘-𝐴))↑2) = 𝐴))
31		fveq2 6774	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (ℜ‘𝑥) = (ℜ‘(i · (√‘-𝐴))))
32	31	breq2d 5086	. . . . . . . 8 ⊢ (𝑥 = (i · (√‘-𝐴)) → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘(i · (√‘-𝐴)))))
33		oveq2 7283	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (i · 𝑥) = (i · (i · (√‘-𝐴))))
34		neleq1 3054	. . . . . . . . 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 1435	. . . . . . 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 3561	. . . . . 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 413	. . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) = -𝐴 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
40	8, 39	sylbid 239	. . 3 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) = 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
41		resqrtcl 14965	. . . . . . . . 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 11003	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (√‘(abs‘𝐴)) ∈ ℂ)
44	43	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (√‘(abs‘𝐴)) ∈ ℂ)
45		addcl 10953	. . . . . . . . 9 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((abs‘𝐴) + 𝐴) ∈ ℂ)
46	2, 45	mpancom 685	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + 𝐴) ∈ ℂ)
47	46	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((abs‘𝐴) + 𝐴) ∈ ℂ)
48		abscl 14990	. . . . . . . . . 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 11003	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (abs‘((abs‘𝐴) + 𝐴)) ∈ ℂ)
51	50	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (abs‘((abs‘𝐴) + 𝐴)) ∈ ℂ)
52	46	abs00ad 15002	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((abs‘((abs‘𝐴) + 𝐴)) = 0 ↔ ((abs‘𝐴) + 𝐴) = 0))
53	52	necon3bid 2988	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((abs‘((abs‘𝐴) + 𝐴)) ≠ 0 ↔ ((abs‘𝐴) + 𝐴) ≠ 0))
54	53	biimpar 478	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (abs‘((abs‘𝐴) + 𝐴)) ≠ 0)
55	47, 51, 54	divcld 11751	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))) ∈ ℂ)
56	44, 55	mulcld 10995	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) ∈ ℂ)
57		eqid 2738	. . . . . 6 ⊢ ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))
58	57	sqreulem 15071	. . . . 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 7282	. . . . . . . 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 2740	. . . . . . 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 6774	. . . . . . . 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 5086	. . . . . . 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 7283	. . . . . . . 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 3054	. . . . . . . 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 1435	. . . . . 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 3561	. . . . 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 413	. . 3 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) ≠ 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
70	40, 69	pm2.61dne 3031	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
71		sqrmo 14963	. 2 ⊢ (𝐴 ∈ ℂ → ∃*𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
72		reu5 3361	. 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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∉ wnel 3049 ∃wrex 3065 ∃!wreu 3066 ∃*wrmo 3067 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 ici 10873 + caddc 10874 · cmul 10876 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 2c2 12028 ℝ⁺crp 12730 ↑cexp 13782 ℜcre 14808 √csqrt 14944 abscabs 14945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947

This theorem is referenced by: sqrtcl 15073 sqrtthlem 15074 eqsqrtd 15079

W3C validator