Step |
Hyp |
Ref |
Expression |
1 |
|
negex |
⊢ - 1 ∈ V |
2 |
1
|
prid1 |
⊢ - 1 ∈ { - 1 , 1 } |
3 |
|
neg1ne0 |
⊢ - 1 ≠ 0 |
4 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
5 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
6 |
|
prssi |
⊢ ( ( - 1 ∈ ℂ ∧ 1 ∈ ℂ ) → { - 1 , 1 } ⊆ ℂ ) |
7 |
4 5 6
|
mp2an |
⊢ { - 1 , 1 } ⊆ ℂ |
8 |
|
elpri |
⊢ ( 𝑥 ∈ { - 1 , 1 } → ( 𝑥 = - 1 ∨ 𝑥 = 1 ) ) |
9 |
7
|
sseli |
⊢ ( 𝑦 ∈ { - 1 , 1 } → 𝑦 ∈ ℂ ) |
10 |
9
|
mulm1d |
⊢ ( 𝑦 ∈ { - 1 , 1 } → ( - 1 · 𝑦 ) = - 𝑦 ) |
11 |
|
elpri |
⊢ ( 𝑦 ∈ { - 1 , 1 } → ( 𝑦 = - 1 ∨ 𝑦 = 1 ) ) |
12 |
|
negeq |
⊢ ( 𝑦 = - 1 → - 𝑦 = - - 1 ) |
13 |
|
negneg1e1 |
⊢ - - 1 = 1 |
14 |
|
1ex |
⊢ 1 ∈ V |
15 |
14
|
prid2 |
⊢ 1 ∈ { - 1 , 1 } |
16 |
13 15
|
eqeltri |
⊢ - - 1 ∈ { - 1 , 1 } |
17 |
12 16
|
eqeltrdi |
⊢ ( 𝑦 = - 1 → - 𝑦 ∈ { - 1 , 1 } ) |
18 |
|
negeq |
⊢ ( 𝑦 = 1 → - 𝑦 = - 1 ) |
19 |
18 2
|
eqeltrdi |
⊢ ( 𝑦 = 1 → - 𝑦 ∈ { - 1 , 1 } ) |
20 |
17 19
|
jaoi |
⊢ ( ( 𝑦 = - 1 ∨ 𝑦 = 1 ) → - 𝑦 ∈ { - 1 , 1 } ) |
21 |
11 20
|
syl |
⊢ ( 𝑦 ∈ { - 1 , 1 } → - 𝑦 ∈ { - 1 , 1 } ) |
22 |
10 21
|
eqeltrd |
⊢ ( 𝑦 ∈ { - 1 , 1 } → ( - 1 · 𝑦 ) ∈ { - 1 , 1 } ) |
23 |
|
oveq1 |
⊢ ( 𝑥 = - 1 → ( 𝑥 · 𝑦 ) = ( - 1 · 𝑦 ) ) |
24 |
23
|
eleq1d |
⊢ ( 𝑥 = - 1 → ( ( 𝑥 · 𝑦 ) ∈ { - 1 , 1 } ↔ ( - 1 · 𝑦 ) ∈ { - 1 , 1 } ) ) |
25 |
22 24
|
syl5ibr |
⊢ ( 𝑥 = - 1 → ( 𝑦 ∈ { - 1 , 1 } → ( 𝑥 · 𝑦 ) ∈ { - 1 , 1 } ) ) |
26 |
9
|
mulid2d |
⊢ ( 𝑦 ∈ { - 1 , 1 } → ( 1 · 𝑦 ) = 𝑦 ) |
27 |
|
id |
⊢ ( 𝑦 ∈ { - 1 , 1 } → 𝑦 ∈ { - 1 , 1 } ) |
28 |
26 27
|
eqeltrd |
⊢ ( 𝑦 ∈ { - 1 , 1 } → ( 1 · 𝑦 ) ∈ { - 1 , 1 } ) |
29 |
|
oveq1 |
⊢ ( 𝑥 = 1 → ( 𝑥 · 𝑦 ) = ( 1 · 𝑦 ) ) |
30 |
29
|
eleq1d |
⊢ ( 𝑥 = 1 → ( ( 𝑥 · 𝑦 ) ∈ { - 1 , 1 } ↔ ( 1 · 𝑦 ) ∈ { - 1 , 1 } ) ) |
31 |
28 30
|
syl5ibr |
⊢ ( 𝑥 = 1 → ( 𝑦 ∈ { - 1 , 1 } → ( 𝑥 · 𝑦 ) ∈ { - 1 , 1 } ) ) |
32 |
25 31
|
jaoi |
⊢ ( ( 𝑥 = - 1 ∨ 𝑥 = 1 ) → ( 𝑦 ∈ { - 1 , 1 } → ( 𝑥 · 𝑦 ) ∈ { - 1 , 1 } ) ) |
33 |
8 32
|
syl |
⊢ ( 𝑥 ∈ { - 1 , 1 } → ( 𝑦 ∈ { - 1 , 1 } → ( 𝑥 · 𝑦 ) ∈ { - 1 , 1 } ) ) |
34 |
33
|
imp |
⊢ ( ( 𝑥 ∈ { - 1 , 1 } ∧ 𝑦 ∈ { - 1 , 1 } ) → ( 𝑥 · 𝑦 ) ∈ { - 1 , 1 } ) |
35 |
|
oveq2 |
⊢ ( 𝑥 = - 1 → ( 1 / 𝑥 ) = ( 1 / - 1 ) ) |
36 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
37 |
|
divneg2 |
⊢ ( ( 1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0 ) → - ( 1 / 1 ) = ( 1 / - 1 ) ) |
38 |
5 5 36 37
|
mp3an |
⊢ - ( 1 / 1 ) = ( 1 / - 1 ) |
39 |
|
1div1e1 |
⊢ ( 1 / 1 ) = 1 |
40 |
39
|
negeqi |
⊢ - ( 1 / 1 ) = - 1 |
41 |
38 40
|
eqtr3i |
⊢ ( 1 / - 1 ) = - 1 |
42 |
41 2
|
eqeltri |
⊢ ( 1 / - 1 ) ∈ { - 1 , 1 } |
43 |
35 42
|
eqeltrdi |
⊢ ( 𝑥 = - 1 → ( 1 / 𝑥 ) ∈ { - 1 , 1 } ) |
44 |
|
oveq2 |
⊢ ( 𝑥 = 1 → ( 1 / 𝑥 ) = ( 1 / 1 ) ) |
45 |
39 15
|
eqeltri |
⊢ ( 1 / 1 ) ∈ { - 1 , 1 } |
46 |
44 45
|
eqeltrdi |
⊢ ( 𝑥 = 1 → ( 1 / 𝑥 ) ∈ { - 1 , 1 } ) |
47 |
43 46
|
jaoi |
⊢ ( ( 𝑥 = - 1 ∨ 𝑥 = 1 ) → ( 1 / 𝑥 ) ∈ { - 1 , 1 } ) |
48 |
8 47
|
syl |
⊢ ( 𝑥 ∈ { - 1 , 1 } → ( 1 / 𝑥 ) ∈ { - 1 , 1 } ) |
49 |
48
|
adantr |
⊢ ( ( 𝑥 ∈ { - 1 , 1 } ∧ 𝑥 ≠ 0 ) → ( 1 / 𝑥 ) ∈ { - 1 , 1 } ) |
50 |
7 34 15 49
|
expcl2lem |
⊢ ( ( - 1 ∈ { - 1 , 1 } ∧ - 1 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( - 1 ↑ 𝑁 ) ∈ { - 1 , 1 } ) |
51 |
2 3 50
|
mp3an12 |
⊢ ( 𝑁 ∈ ℤ → ( - 1 ↑ 𝑁 ) ∈ { - 1 , 1 } ) |