Step |
Hyp |
Ref |
Expression |
1 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
2 |
|
1re |
⊢ 1 ∈ ℝ |
3 |
|
0re |
⊢ 0 ∈ ℝ |
4 |
2 3
|
lttri2i |
⊢ ( 1 ≠ 0 ↔ ( 1 < 0 ∨ 0 < 1 ) ) |
5 |
1 4
|
mpbi |
⊢ ( 1 < 0 ∨ 0 < 1 ) |
6 |
|
rernegcl |
⊢ ( 1 ∈ ℝ → ( 0 −ℝ 1 ) ∈ ℝ ) |
7 |
2 6
|
mp1i |
⊢ ( 1 < 0 → ( 0 −ℝ 1 ) ∈ ℝ ) |
8 |
|
relt0neg1 |
⊢ ( 1 ∈ ℝ → ( 1 < 0 ↔ 0 < ( 0 −ℝ 1 ) ) ) |
9 |
2 8
|
ax-mp |
⊢ ( 1 < 0 ↔ 0 < ( 0 −ℝ 1 ) ) |
10 |
9
|
biimpi |
⊢ ( 1 < 0 → 0 < ( 0 −ℝ 1 ) ) |
11 |
7 7 10 10
|
mulgt0d |
⊢ ( 1 < 0 → 0 < ( ( 0 −ℝ 1 ) · ( 0 −ℝ 1 ) ) ) |
12 |
|
1red |
⊢ ( 1 ∈ ℝ → 1 ∈ ℝ ) |
13 |
6 12
|
remulneg2d |
⊢ ( 1 ∈ ℝ → ( ( 0 −ℝ 1 ) · ( 0 −ℝ 1 ) ) = ( 0 −ℝ ( ( 0 −ℝ 1 ) · 1 ) ) ) |
14 |
|
ax-1rid |
⊢ ( ( 0 −ℝ 1 ) ∈ ℝ → ( ( 0 −ℝ 1 ) · 1 ) = ( 0 −ℝ 1 ) ) |
15 |
6 14
|
syl |
⊢ ( 1 ∈ ℝ → ( ( 0 −ℝ 1 ) · 1 ) = ( 0 −ℝ 1 ) ) |
16 |
15
|
oveq2d |
⊢ ( 1 ∈ ℝ → ( 0 −ℝ ( ( 0 −ℝ 1 ) · 1 ) ) = ( 0 −ℝ ( 0 −ℝ 1 ) ) ) |
17 |
|
renegneg |
⊢ ( 1 ∈ ℝ → ( 0 −ℝ ( 0 −ℝ 1 ) ) = 1 ) |
18 |
13 16 17
|
3eqtrd |
⊢ ( 1 ∈ ℝ → ( ( 0 −ℝ 1 ) · ( 0 −ℝ 1 ) ) = 1 ) |
19 |
2 18
|
ax-mp |
⊢ ( ( 0 −ℝ 1 ) · ( 0 −ℝ 1 ) ) = 1 |
20 |
11 19
|
breqtrdi |
⊢ ( 1 < 0 → 0 < 1 ) |
21 |
|
id |
⊢ ( 0 < 1 → 0 < 1 ) |
22 |
20 21
|
jaoi |
⊢ ( ( 1 < 0 ∨ 0 < 1 ) → 0 < 1 ) |
23 |
5 22
|
ax-mp |
⊢ 0 < 1 |