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Theorem sqreu 15000

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 14918	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
2	1	recnd 10934	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ)
3		subneg 11200	. . . . . . 7 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((abs‘𝐴) − -𝐴) = ((abs‘𝐴) + 𝐴))
4	2, 3	mpancom 684	. . . . . 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 11151	. . . . . 6 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
7	2, 6	subeq0ad 11272	. . . . 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 10861	. . . . . . 7 ⊢ i ∈ ℂ
10		absge0 14927	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴))
11	1, 10	jca 511	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
12		eleq1 2826	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = -𝐴 → ((abs‘𝐴) ∈ ℝ ↔ -𝐴 ∈ ℝ))
13		breq2 5074	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) = -𝐴 → (0 ≤ (abs‘𝐴) ↔ 0 ≤ -𝐴))
14	12, 13	anbi12d 630	. . . . . . . . . . 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 407	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴))
17		resqrtcl 14893	. . . . . . . . 9 ⊢ ((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) → (√‘-𝐴) ∈ ℝ)
18	16, 17	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (√‘-𝐴) ∈ ℝ)
19	18	recnd 10934	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (√‘-𝐴) ∈ ℂ)
20		mulcl 10886	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ (√‘-𝐴) ∈ ℂ) → (i · (√‘-𝐴)) ∈ ℂ)
21	9, 19, 20	sylancr 586	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = -𝐴) → (i · (√‘-𝐴)) ∈ ℂ)
22		sqrtneglem 14906	. . . . . . . 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 11201	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴)
25	24	adantr 480	. . . . . . . . 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 1438	. . . . . . 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 7262	. . . . . . . . 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 6756	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (ℜ‘𝑥) = (ℜ‘(i · (√‘-𝐴))))
32	31	breq2d 5082	. . . . . . . 8 ⊢ (𝑥 = (i · (√‘-𝐴)) → (0 ≤ (ℜ‘𝑥) ↔ 0 ≤ (ℜ‘(i · (√‘-𝐴)))))
33		oveq2 7263	. . . . . . . . 9 ⊢ (𝑥 = (i · (√‘-𝐴)) → (i · 𝑥) = (i · (i · (√‘-𝐴))))
34		neleq1 3053	. . . . . . . . 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 1434	. . . . . . 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 3552	. . . . . 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 583	. . . . 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 239	. . 3 ⊢ (𝐴 ∈ ℂ → (((abs‘𝐴) + 𝐴) = 0 → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺)))
41		resqrtcl 14893	. . . . . . . . 9 ⊢ (((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) → (√‘(abs‘𝐴)) ∈ ℝ)
42	1, 10, 41	syl2anc 583	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (√‘(abs‘𝐴)) ∈ ℝ)
43	42	recnd 10934	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (√‘(abs‘𝐴)) ∈ ℂ)
44	43	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (√‘(abs‘𝐴)) ∈ ℂ)
45		addcl 10884	. . . . . . . . 9 ⊢ (((abs‘𝐴) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((abs‘𝐴) + 𝐴) ∈ ℂ)
46	2, 45	mpancom 684	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + 𝐴) ∈ ℂ)
47	46	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → ((abs‘𝐴) + 𝐴) ∈ ℂ)
48		abscl 14918	. . . . . . . . . 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 10934	. . . . . . . 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 14930	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((abs‘((abs‘𝐴) + 𝐴)) = 0 ↔ ((abs‘𝐴) + 𝐴) = 0))
53	52	necon3bid 2987	. . . . . . . 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 11681	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ((abs‘𝐴) + 𝐴) ≠ 0) → (((abs‘𝐴) + 𝐴) / (abs‘((abs‘𝐴) + 𝐴))) ∈ ℂ)
56	44, 55	mulcld 10926	. . . . 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 14999	. . . . 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 7262	. . . . . . . 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 6756	. . . . . . . 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 5082	. . . . . . 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 7263	. . . . . . . 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 3053	. . . . . . . 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 1434	. . . . . 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 3552	. . . . 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 583	. . . 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 3030	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
71		sqrmo 14891	. 2 ⊢ (𝐴 ∈ ℂ → ∃*𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ⁺))
72		reu5 3351	. 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 582	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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∉ wnel 3048 ∃wrex 3064 ∃!wreu 3065 ∃*wrmo 3066 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 ici 10804 + caddc 10805 · cmul 10807 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 2c2 11958 ℝ⁺crp 12659 ↑cexp 13710 ℜcre 14736 √csqrt 14872 abscabs 14873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875

This theorem is referenced by: sqrtcl 15001 sqrtthlem 15002 eqsqrtd 15007

W3C validator