Metamath Proof Explorer


Theorem sqrtnegnre

Description: The square root of a negative number is not a real number. (Contributed by AV, 28-Feb-2023)

Ref Expression
Assertion sqrtnegnre ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ( √ ‘ 𝑋 ) ∉ ℝ )

Proof

Step Hyp Ref Expression
1 recn ( 𝑋 ∈ ℝ → 𝑋 ∈ ℂ )
2 1 negnegd ( 𝑋 ∈ ℝ → - - 𝑋 = 𝑋 )
3 2 adantr ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → - - 𝑋 = 𝑋 )
4 3 eqcomd ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → 𝑋 = - - 𝑋 )
5 4 fveq2d ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ( √ ‘ 𝑋 ) = ( √ ‘ - - 𝑋 ) )
6 simpl ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → 𝑋 ∈ ℝ )
7 6 renegcld ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → - 𝑋 ∈ ℝ )
8 0re 0 ∈ ℝ
9 ltle ( ( 𝑋 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝑋 < 0 → 𝑋 ≤ 0 ) )
10 8 9 mpan2 ( 𝑋 ∈ ℝ → ( 𝑋 < 0 → 𝑋 ≤ 0 ) )
11 10 imp ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → 𝑋 ≤ 0 )
12 le0neg1 ( 𝑋 ∈ ℝ → ( 𝑋 ≤ 0 ↔ 0 ≤ - 𝑋 ) )
13 12 adantr ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ( 𝑋 ≤ 0 ↔ 0 ≤ - 𝑋 ) )
14 11 13 mpbid ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → 0 ≤ - 𝑋 )
15 7 14 sqrtnegd ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ( √ ‘ - - 𝑋 ) = ( i · ( √ ‘ - 𝑋 ) ) )
16 5 15 eqtrd ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ( √ ‘ 𝑋 ) = ( i · ( √ ‘ - 𝑋 ) ) )
17 ax-icn i ∈ ℂ
18 17 a1i ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → i ∈ ℂ )
19 1 adantr ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → 𝑋 ∈ ℂ )
20 19 negcld ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → - 𝑋 ∈ ℂ )
21 20 sqrtcld ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ( √ ‘ - 𝑋 ) ∈ ℂ )
22 18 21 mulcomd ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ( i · ( √ ‘ - 𝑋 ) ) = ( ( √ ‘ - 𝑋 ) · i ) )
23 7 14 resqrtcld ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ( √ ‘ - 𝑋 ) ∈ ℝ )
24 inelr ¬ i ∈ ℝ
25 24 a1i ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ¬ i ∈ ℝ )
26 18 25 eldifd ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → i ∈ ( ℂ ∖ ℝ ) )
27 lt0neg1 ( 𝑋 ∈ ℝ → ( 𝑋 < 0 ↔ 0 < - 𝑋 ) )
28 8 a1i ( 𝑋 ∈ ℝ → 0 ∈ ℝ )
29 ltne ( ( 0 ∈ ℝ ∧ 0 < - 𝑋 ) → - 𝑋 ≠ 0 )
30 28 29 sylan ( ( 𝑋 ∈ ℝ ∧ 0 < - 𝑋 ) → - 𝑋 ≠ 0 )
31 simpl ( ( 𝑋 ∈ ℝ ∧ 0 < - 𝑋 ) → 𝑋 ∈ ℝ )
32 31 renegcld ( ( 𝑋 ∈ ℝ ∧ 0 < - 𝑋 ) → - 𝑋 ∈ ℝ )
33 10 27 12 3imtr3d ( 𝑋 ∈ ℝ → ( 0 < - 𝑋 → 0 ≤ - 𝑋 ) )
34 33 imp ( ( 𝑋 ∈ ℝ ∧ 0 < - 𝑋 ) → 0 ≤ - 𝑋 )
35 sqrt00 ( ( - 𝑋 ∈ ℝ ∧ 0 ≤ - 𝑋 ) → ( ( √ ‘ - 𝑋 ) = 0 ↔ - 𝑋 = 0 ) )
36 32 34 35 syl2anc ( ( 𝑋 ∈ ℝ ∧ 0 < - 𝑋 ) → ( ( √ ‘ - 𝑋 ) = 0 ↔ - 𝑋 = 0 ) )
37 36 bicomd ( ( 𝑋 ∈ ℝ ∧ 0 < - 𝑋 ) → ( - 𝑋 = 0 ↔ ( √ ‘ - 𝑋 ) = 0 ) )
38 37 necon3bid ( ( 𝑋 ∈ ℝ ∧ 0 < - 𝑋 ) → ( - 𝑋 ≠ 0 ↔ ( √ ‘ - 𝑋 ) ≠ 0 ) )
39 30 38 mpbid ( ( 𝑋 ∈ ℝ ∧ 0 < - 𝑋 ) → ( √ ‘ - 𝑋 ) ≠ 0 )
40 39 ex ( 𝑋 ∈ ℝ → ( 0 < - 𝑋 → ( √ ‘ - 𝑋 ) ≠ 0 ) )
41 27 40 sylbid ( 𝑋 ∈ ℝ → ( 𝑋 < 0 → ( √ ‘ - 𝑋 ) ≠ 0 ) )
42 41 imp ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ( √ ‘ - 𝑋 ) ≠ 0 )
43 23 26 42 recnmulnred ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ( ( √ ‘ - 𝑋 ) · i ) ∉ ℝ )
44 df-nel ( ( ( √ ‘ - 𝑋 ) · i ) ∉ ℝ ↔ ¬ ( ( √ ‘ - 𝑋 ) · i ) ∈ ℝ )
45 43 44 sylib ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ¬ ( ( √ ‘ - 𝑋 ) · i ) ∈ ℝ )
46 22 45 eqneltrd ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ¬ ( i · ( √ ‘ - 𝑋 ) ) ∈ ℝ )
47 16 46 eqneltrd ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ¬ ( √ ‘ 𝑋 ) ∈ ℝ )
48 df-nel ( ( √ ‘ 𝑋 ) ∉ ℝ ↔ ¬ ( √ ‘ 𝑋 ) ∈ ℝ )
49 47 48 sylibr ( ( 𝑋 ∈ ℝ ∧ 𝑋 < 0 ) → ( √ ‘ 𝑋 ) ∉ ℝ )