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 ) |