Step |
Hyp |
Ref |
Expression |
1 |
|
reixi |
⊢ ( i · i ) = ( 0 −ℝ 1 ) |
2 |
1 1
|
oveq12i |
⊢ ( ( i · i ) · ( i · i ) ) = ( ( 0 −ℝ 1 ) · ( 0 −ℝ 1 ) ) |
3 |
|
1re |
⊢ 1 ∈ ℝ |
4 |
|
rernegcl |
⊢ ( 1 ∈ ℝ → ( 0 −ℝ 1 ) ∈ ℝ ) |
5 |
|
1red |
⊢ ( 1 ∈ ℝ → 1 ∈ ℝ ) |
6 |
4 5
|
remulneg2d |
⊢ ( 1 ∈ ℝ → ( ( 0 −ℝ 1 ) · ( 0 −ℝ 1 ) ) = ( 0 −ℝ ( ( 0 −ℝ 1 ) · 1 ) ) ) |
7 |
|
ax-1rid |
⊢ ( ( 0 −ℝ 1 ) ∈ ℝ → ( ( 0 −ℝ 1 ) · 1 ) = ( 0 −ℝ 1 ) ) |
8 |
4 7
|
syl |
⊢ ( 1 ∈ ℝ → ( ( 0 −ℝ 1 ) · 1 ) = ( 0 −ℝ 1 ) ) |
9 |
8
|
oveq2d |
⊢ ( 1 ∈ ℝ → ( 0 −ℝ ( ( 0 −ℝ 1 ) · 1 ) ) = ( 0 −ℝ ( 0 −ℝ 1 ) ) ) |
10 |
|
renegneg |
⊢ ( 1 ∈ ℝ → ( 0 −ℝ ( 0 −ℝ 1 ) ) = 1 ) |
11 |
6 9 10
|
3eqtrd |
⊢ ( 1 ∈ ℝ → ( ( 0 −ℝ 1 ) · ( 0 −ℝ 1 ) ) = 1 ) |
12 |
3 11
|
ax-mp |
⊢ ( ( 0 −ℝ 1 ) · ( 0 −ℝ 1 ) ) = 1 |
13 |
2 12
|
eqtri |
⊢ ( ( i · i ) · ( i · i ) ) = 1 |