Step |
Hyp |
Ref |
Expression |
1 |
|
it1ei |
⊢ ( i · 1 ) = i |
2 |
1
|
eqcomi |
⊢ i = ( i · 1 ) |
3 |
|
reixi |
⊢ ( i · i ) = ( 0 −ℝ 1 ) |
4 |
3
|
oveq2i |
⊢ ( i · ( i · i ) ) = ( i · ( 0 −ℝ 1 ) ) |
5 |
2 4
|
oveq12i |
⊢ ( i + ( i · ( i · i ) ) ) = ( ( i · 1 ) + ( i · ( 0 −ℝ 1 ) ) ) |
6 |
|
ax-icn |
⊢ i ∈ ℂ |
7 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
8 |
|
1re |
⊢ 1 ∈ ℝ |
9 |
|
rernegcl |
⊢ ( 1 ∈ ℝ → ( 0 −ℝ 1 ) ∈ ℝ ) |
10 |
8 9
|
ax-mp |
⊢ ( 0 −ℝ 1 ) ∈ ℝ |
11 |
10
|
recni |
⊢ ( 0 −ℝ 1 ) ∈ ℂ |
12 |
6 7 11
|
adddii |
⊢ ( i · ( 1 + ( 0 −ℝ 1 ) ) ) = ( ( i · 1 ) + ( i · ( 0 −ℝ 1 ) ) ) |
13 |
|
renegid |
⊢ ( 1 ∈ ℝ → ( 1 + ( 0 −ℝ 1 ) ) = 0 ) |
14 |
8 13
|
ax-mp |
⊢ ( 1 + ( 0 −ℝ 1 ) ) = 0 |
15 |
14
|
oveq2i |
⊢ ( i · ( 1 + ( 0 −ℝ 1 ) ) ) = ( i · 0 ) |
16 |
|
sn-it0e0 |
⊢ ( i · 0 ) = 0 |
17 |
15 16
|
eqtri |
⊢ ( i · ( 1 + ( 0 −ℝ 1 ) ) ) = 0 |
18 |
5 12 17
|
3eqtr2i |
⊢ ( i + ( i · ( i · i ) ) ) = 0 |