Theorem sqrt0

Description: Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013)

	Ref	Expression
Assertion	sqrt0	⊢ ( √ ‘ 0 ) = 0

Proof

Step	Hyp	Ref	Expression
1		0cn	⊢ 0 ∈ ℂ
2		sqrtval	⊢ ( 0 ∈ ℂ → ( √ ‘ 0 ) = ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) )
3	1 2	ax-mp	⊢ ( √ ‘ 0 ) = ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) )
4		id	⊢ ( 0 ∈ ℂ → 0 ∈ ℂ )
5		sqr0lem	⊢ ( ( 𝑥 ∈ ℂ ∧ ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) ↔ 𝑥 = 0 )
6	5	biimpi	⊢ ( ( 𝑥 ∈ ℂ ∧ ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) → 𝑥 = 0 )
7	6	ex	⊢ ( 𝑥 ∈ ℂ → ( ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) → 𝑥 = 0 ) )
8		simpr	⊢ ( ( 𝑥 ∈ ℂ ∧ ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) → ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) )
9	5 8	sylbir	⊢ ( 𝑥 = 0 → ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) )
10	7 9	impbid1	⊢ ( 𝑥 ∈ ℂ → ( ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ↔ 𝑥 = 0 ) )
11	10	adantl	⊢ ( ( 0 ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ↔ 𝑥 = 0 ) )
12	4 11	riota5	⊢ ( 0 ∈ ℂ → ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) = 0 )
13	1 12	ax-mp	⊢ ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 0 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) = 0
14	3 13	eqtri	⊢ ( √ ‘ 0 ) = 0