Step |
Hyp |
Ref |
Expression |
1 |
|
0ne1 |
⊢ 0 ≠ 1 |
2 |
|
ax-icn |
⊢ i ∈ ℂ |
3 |
2 2
|
mulcli |
⊢ ( i · i ) ∈ ℂ |
4 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
5 |
3 4 4
|
addassi |
⊢ ( ( ( i · i ) + 1 ) + 1 ) = ( ( i · i ) + ( 1 + 1 ) ) |
6 |
5
|
a1i |
⊢ ( ( 0 = ( 0 + 0 ) ∧ 1 = ( 1 + 1 ) ) → ( ( ( i · i ) + 1 ) + 1 ) = ( ( i · i ) + ( 1 + 1 ) ) ) |
7 |
|
simpr |
⊢ ( ( 0 = ( 0 + 0 ) ∧ 1 = ( 1 + 1 ) ) → 1 = ( 1 + 1 ) ) |
8 |
7
|
oveq2d |
⊢ ( ( 0 = ( 0 + 0 ) ∧ 1 = ( 1 + 1 ) ) → ( ( i · i ) + 1 ) = ( ( i · i ) + ( 1 + 1 ) ) ) |
9 |
|
ax-i2m1 |
⊢ ( ( i · i ) + 1 ) = 0 |
10 |
9
|
a1i |
⊢ ( ( 0 = ( 0 + 0 ) ∧ 1 = ( 1 + 1 ) ) → ( ( i · i ) + 1 ) = 0 ) |
11 |
6 8 10
|
3eqtr2rd |
⊢ ( ( 0 = ( 0 + 0 ) ∧ 1 = ( 1 + 1 ) ) → 0 = ( ( ( i · i ) + 1 ) + 1 ) ) |
12 |
|
simpl |
⊢ ( ( 0 = ( 0 + 0 ) ∧ 1 = ( 1 + 1 ) ) → 0 = ( 0 + 0 ) ) |
13 |
10
|
oveq1d |
⊢ ( ( 0 = ( 0 + 0 ) ∧ 1 = ( 1 + 1 ) ) → ( ( ( i · i ) + 1 ) + 1 ) = ( 0 + 1 ) ) |
14 |
11 12 13
|
3eqtr3d |
⊢ ( ( 0 = ( 0 + 0 ) ∧ 1 = ( 1 + 1 ) ) → ( 0 + 0 ) = ( 0 + 1 ) ) |
15 |
|
0red |
⊢ ( ( 0 = ( 0 + 0 ) ∧ 1 = ( 1 + 1 ) ) → 0 ∈ ℝ ) |
16 |
|
1red |
⊢ ( ( 0 = ( 0 + 0 ) ∧ 1 = ( 1 + 1 ) ) → 1 ∈ ℝ ) |
17 |
|
readdcan |
⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( 0 + 0 ) = ( 0 + 1 ) ↔ 0 = 1 ) ) |
18 |
15 16 15 17
|
syl3anc |
⊢ ( ( 0 = ( 0 + 0 ) ∧ 1 = ( 1 + 1 ) ) → ( ( 0 + 0 ) = ( 0 + 1 ) ↔ 0 = 1 ) ) |
19 |
14 18
|
mpbid |
⊢ ( ( 0 = ( 0 + 0 ) ∧ 1 = ( 1 + 1 ) ) → 0 = 1 ) |
20 |
19
|
ex |
⊢ ( 0 = ( 0 + 0 ) → ( 1 = ( 1 + 1 ) → 0 = 1 ) ) |
21 |
20
|
necon3d |
⊢ ( 0 = ( 0 + 0 ) → ( 0 ≠ 1 → 1 ≠ ( 1 + 1 ) ) ) |
22 |
1 21
|
mpi |
⊢ ( 0 = ( 0 + 0 ) → 1 ≠ ( 1 + 1 ) ) |
23 |
|
oveq2 |
⊢ ( 1 = ( 1 + 1 ) → ( 0 · 1 ) = ( 0 · ( 1 + 1 ) ) ) |
24 |
|
0re |
⊢ 0 ∈ ℝ |
25 |
|
ax-1rid |
⊢ ( 0 ∈ ℝ → ( 0 · 1 ) = 0 ) |
26 |
24 25
|
ax-mp |
⊢ ( 0 · 1 ) = 0 |
27 |
|
0cn |
⊢ 0 ∈ ℂ |
28 |
27 4 4
|
adddii |
⊢ ( 0 · ( 1 + 1 ) ) = ( ( 0 · 1 ) + ( 0 · 1 ) ) |
29 |
26 26
|
oveq12i |
⊢ ( ( 0 · 1 ) + ( 0 · 1 ) ) = ( 0 + 0 ) |
30 |
28 29
|
eqtri |
⊢ ( 0 · ( 1 + 1 ) ) = ( 0 + 0 ) |
31 |
23 26 30
|
3eqtr3g |
⊢ ( 1 = ( 1 + 1 ) → 0 = ( 0 + 0 ) ) |
32 |
31
|
necon3i |
⊢ ( 0 ≠ ( 0 + 0 ) → 1 ≠ ( 1 + 1 ) ) |
33 |
22 32
|
pm2.61ine |
⊢ 1 ≠ ( 1 + 1 ) |
34 |
|
df-2 |
⊢ 2 = ( 1 + 1 ) |
35 |
33 34
|
neeqtrri |
⊢ 1 ≠ 2 |