sqreu - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > sqreu		Structured version Visualization version GIF version

Theorem sqreu 14100

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 14018	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
2	1	recnd 10068	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ)
3		subneg 10330	. . . . . . 7 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((abs‘𝐴) − -𝐴) = ((abs‘𝐴) + 𝐴))
4	2, 3	mpancom 703	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) − -𝐴) = ((abs‘𝐴) + 𝐴))
5	4	eqeq1d 2624	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) − -𝐴) = 0 ↔ ((abs‘𝐴) + 𝐴) = 0))
6		negcl 10281	. . . . . 6 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
7	2, 6	subeq0ad 10402	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) − -𝐴) = 0 ↔ (abs‘𝐴) = -𝐴))
8	5, 7	bitr3d 270	. . . 4 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) = 0 ↔ (abs‘𝐴) = -𝐴))
9		ax-icn 9995	. . . . . . 7 ⊢ i ∈ ℂ
10		absge0 14027	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
11	1, 10	jca 554	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
12		eleq1 2689	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = -𝐴 → ((abs‘𝐴) ∈ ℝ ↔ -𝐴 ∈ ℝ))
13		breq2 4657	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = -𝐴 → (0 ≤ (abs‘𝐴) ↔ 0 ≤ -𝐴))
14	12, 13	anbi12d 747	. . . . . . . . . . 11 ⊢ ((abs‘𝐴) = -𝐴 → (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) ↔ (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴)))
15	11, 14	syl5ib 234	. . . . . . . . . 10 ⊢ ((abs‘𝐴) = -𝐴 → (𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴)))
16	15	impcom 446	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴))
17		resqrtcl 13994	. . . . . . . . 9 ⊢ ((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) → (√‘-𝐴) ∈ ℝ)
18	16, 17	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (√‘-𝐴) ∈ ℝ)
19	18	recnd 10068	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (√‘-𝐴) ∈ ℂ)
20		mulcl 10020	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ (√‘-𝐴) ∈ ℂ) → (i · (√‘-𝐴)) ∈ ℂ)
21	9, 19, 20	sylancr 695	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (i · (√‘-𝐴)) ∈ ℂ)
22		sqrtneglem 14007	. . . . . . . 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 10331	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴)
25	24	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → --𝐴 = 𝐴)
26	25	eqeq2d 2632	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (((i · (√‘-𝐴))↑2) = --𝐴 ↔ ((i · (√‘-𝐴))↑2) = 𝐴))
27	26	3anbi1d 1403	. . . . . . 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 222	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (((i · (√‘-𝐴))↑2) = 𝐴 ∧ 0 ≤ (ℜ‘(i · (√‘-𝐴))) ∧ (i · (i · (√‘-𝐴))) ∉ ℝ⁺))
29		oveq1 6657	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (𝑥↑2) = ((i · (√‘-𝐴))↑2))
30	29	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑥 = (i · (√‘-𝐴)) → ((𝑥↑2) = 𝐴 ↔ ((i · (√‘-𝐴))↑2) = 𝐴))
31		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (ℜ‘𝑥) = (ℜ‘(i · (√‘-𝐴))))
32	31	breq2d 4665	. . . . . . . 8 ⊢ (𝑥 = (i · (√‘-𝐴)) → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘(i · (√‘-𝐴)))))
33		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (i · 𝑥) = (i · (i · (√‘-𝐴))))
34		neleq1 2902	. . . . . . . . 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 1399	. . . . . . 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 3309	. . . . . 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 693	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
39	38	ex 450	. . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) = -𝐴 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
40	8, 39	sylbid 230	. . 3 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) = 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
41		resqrtcl 13994	. . . . . . . . 9 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) → (√‘(abs‘𝐴)) ∈ ℝ)
42	1, 10, 41	syl2anc 693	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (√‘(abs‘𝐴)) ∈ ℝ)
43	42	recnd 10068	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (√‘(abs‘𝐴)) ∈ ℂ)
44	43	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (√‘(abs‘𝐴)) ∈ ℂ)
45		addcl 10018	. . . . . . . . 9 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((abs‘𝐴) + 𝐴) ∈ ℂ)
46	2, 45	mpancom 703	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + 𝐴) ∈ ℂ)
47	46	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((abs‘𝐴) + 𝐴) ∈ ℂ)
48		abscl 14018	. . . . . . . . . 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 10068	. . . . . . . 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 14030	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((abs‘((abs‘𝐴) + 𝐴)) = 0 ↔ ((abs‘𝐴) + 𝐴) = 0))
53	52	necon3bid 2838	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((abs‘((abs‘𝐴) + 𝐴)) ≠ 0 ↔ ((abs‘𝐴) + 𝐴) ≠ 0))
54	53	biimpar 502	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (abs‘((abs‘𝐴) + 𝐴)) ≠ 0)
55	47, 51, 54	divcld 10801	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))) ∈ ℂ)
56	44, 55	mulcld 10060	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) ∈ ℂ)
57		eqid 2622	. . . . . 6 ⊢ ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴)))) = ((√‘(abs‘𝐴)) · (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))))
58	57	sqreulem 14099	. . . . 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 6657	. . . . . . . 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 2624	. . . . . . 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 6191	. . . . . . . 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 4665	. . . . . . 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 6658	. . . . . . . 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 2902	. . . . . . . 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 1399	. . . . . 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 3309	. . . . 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 693	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
69	68	ex 450	. . 3 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) ≠ 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
70	40, 69	pm2.61dne 2880	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
71		sqrmo 13992	. 2 ⊢ (𝐴 ∈ ℂ → ∃*𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
72		reu5 3159	. 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 698	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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∉ wnel 2897 ∃wrex 2913 ∃!wreu 2914 ∃*wrmo 2915 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 ici 9938 + caddc 9939 · cmul 9941 ≤ cle 10075 − cmin 10266 -cneg 10267 / cdiv 10684 2c2 11070 ℝ⁺crp 11832 ↑cexp 12860 ℜcre 13837 √csqrt 13973 abscabs 13974

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976

This theorem is referenced by: sqrtcl 14101 sqrtthlem 14102 eqsqrtd 14107

W3C validator