| Step |
Hyp |
Ref |
Expression |
| 1 |
|
gpg5nbgrvtx03starlem1.j |
⊢ 𝐽 = ( 1 ..^ ( ⌈ ‘ ( 𝑁 / 2 ) ) ) |
| 2 |
|
gpg5nbgrvtx03starlem1.g |
⊢ 𝐺 = ( 𝑁 gPetersenGr 𝐾 ) |
| 3 |
|
gpg5nbgrvtx03starlem1.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
| 4 |
|
gpg5nbgrvtx03starlem1.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
| 5 |
|
opex |
⊢ ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ∈ V |
| 6 |
|
opex |
⊢ ⟨ 1 , 𝑋 ⟩ ∈ V |
| 7 |
5 6
|
pm3.2i |
⊢ ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ∈ V ∧ ⟨ 1 , 𝑋 ⟩ ∈ V ) |
| 8 |
|
opex |
⊢ ⟨ 0 , 𝑥 ⟩ ∈ V |
| 9 |
|
opex |
⊢ ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ ∈ V |
| 10 |
8 9
|
pm3.2i |
⊢ ( ⟨ 0 , 𝑥 ⟩ ∈ V ∧ ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ ∈ V ) |
| 11 |
7 10
|
pm3.2i |
⊢ ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ∈ V ∧ ⟨ 1 , 𝑋 ⟩ ∈ V ) ∧ ( ⟨ 0 , 𝑥 ⟩ ∈ V ∧ ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ ∈ V ) ) |
| 12 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
| 13 |
12
|
orci |
⊢ ( 1 ≠ 0 ∨ 𝑋 ≠ 𝑥 ) |
| 14 |
13
|
a1i |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( 1 ≠ 0 ∨ 𝑋 ≠ 𝑥 ) ) |
| 15 |
|
1ex |
⊢ 1 ∈ V |
| 16 |
15
|
a1i |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) → 1 ∈ V ) |
| 17 |
|
simp3 |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) → 𝑋 ∈ 𝑊 ) |
| 18 |
16 17
|
jca |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) → ( 1 ∈ V ∧ 𝑋 ∈ 𝑊 ) ) |
| 19 |
18
|
adantr |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( 1 ∈ V ∧ 𝑋 ∈ 𝑊 ) ) |
| 20 |
|
opthneg |
⊢ ( ( 1 ∈ V ∧ 𝑋 ∈ 𝑊 ) → ( ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ↔ ( 1 ≠ 0 ∨ 𝑋 ≠ 𝑥 ) ) ) |
| 21 |
19 20
|
syl |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ↔ ( 1 ≠ 0 ∨ 𝑋 ≠ 𝑥 ) ) ) |
| 22 |
14 21
|
mpbird |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ) |
| 23 |
12
|
a1i |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → 1 ≠ 0 ) |
| 24 |
23
|
orcd |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( 1 ≠ 0 ∨ 𝑋 ≠ ( ( 𝑥 + 1 ) mod 𝑁 ) ) ) |
| 25 |
|
opthneg |
⊢ ( ( 1 ∈ V ∧ 𝑋 ∈ 𝑊 ) → ( ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ ↔ ( 1 ≠ 0 ∨ 𝑋 ≠ ( ( 𝑥 + 1 ) mod 𝑁 ) ) ) ) |
| 26 |
19 25
|
syl |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ ↔ ( 1 ≠ 0 ∨ 𝑋 ≠ ( ( 𝑥 + 1 ) mod 𝑁 ) ) ) ) |
| 27 |
24 26
|
mpbird |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ ) |
| 28 |
22 27
|
jca |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ∧ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ ) ) |
| 29 |
28
|
olcd |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ∧ ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ ) ∨ ( ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ∧ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ ) ) ) |
| 30 |
|
prneimg |
⊢ ( ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ∈ V ∧ ⟨ 1 , 𝑋 ⟩ ∈ V ) ∧ ( ⟨ 0 , 𝑥 ⟩ ∈ V ∧ ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ ∈ V ) ) → ( ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ∧ ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ ) ∨ ( ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ∧ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ ) ) → { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ) ) |
| 31 |
11 29 30
|
mpsyl |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ) |
| 32 |
|
eleq1 |
⊢ ( 𝑋 = 𝑥 → ( 𝑋 ∈ ( 0 ..^ 𝑁 ) ↔ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) ) |
| 33 |
32
|
adantr |
⊢ ( ( 𝑋 = 𝑥 ∧ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ) → ( 𝑋 ∈ ( 0 ..^ 𝑁 ) ↔ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) ) |
| 34 |
|
uzuzle23 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → 𝑁 ∈ ( ℤ≥ ‘ 2 ) ) |
| 35 |
34
|
3ad2ant1 |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) → 𝑁 ∈ ( ℤ≥ ‘ 2 ) ) |
| 36 |
|
p1modne |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑋 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑋 ) |
| 37 |
35 36
|
sylan |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑋 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑋 ) |
| 38 |
37
|
ex |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) → ( 𝑋 ∈ ( 0 ..^ 𝑁 ) → ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑋 ) ) |
| 39 |
38
|
adantl |
⊢ ( ( 𝑋 = 𝑥 ∧ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ) → ( 𝑋 ∈ ( 0 ..^ 𝑁 ) → ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑋 ) ) |
| 40 |
33 39
|
sylbird |
⊢ ( ( 𝑋 = 𝑥 ∧ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ) → ( 𝑥 ∈ ( 0 ..^ 𝑁 ) → ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑋 ) ) |
| 41 |
40
|
impr |
⊢ ( ( 𝑋 = 𝑥 ∧ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) ) → ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑋 ) |
| 42 |
|
neeq2 |
⊢ ( 𝑋 = 𝑥 → ( ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑋 ↔ ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ) ) |
| 43 |
42
|
adantr |
⊢ ( ( 𝑋 = 𝑥 ∧ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) ) → ( ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑋 ↔ ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ) ) |
| 44 |
41 43
|
mpbid |
⊢ ( ( 𝑋 = 𝑥 ∧ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) ) → ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ) |
| 45 |
44
|
orcd |
⊢ ( ( 𝑋 = 𝑥 ∧ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) ) → ( ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ∨ 𝑋 ≠ 𝑥 ) ) |
| 46 |
45
|
ex |
⊢ ( 𝑋 = 𝑥 → ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ∨ 𝑋 ≠ 𝑥 ) ) ) |
| 47 |
|
olc |
⊢ ( 𝑋 ≠ 𝑥 → ( ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ∨ 𝑋 ≠ 𝑥 ) ) |
| 48 |
47
|
a1d |
⊢ ( 𝑋 ≠ 𝑥 → ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ∨ 𝑋 ≠ 𝑥 ) ) ) |
| 49 |
46 48
|
pm2.61ine |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ∨ 𝑋 ≠ 𝑥 ) ) |
| 50 |
|
c0ex |
⊢ 0 ∈ V |
| 51 |
|
ovex |
⊢ ( ( 𝑋 + 1 ) mod 𝑁 ) ∈ V |
| 52 |
50 51
|
opthne |
⊢ ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ↔ ( 0 ≠ 0 ∨ ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ) ) |
| 53 |
|
neirr |
⊢ ¬ 0 ≠ 0 |
| 54 |
53
|
biorfi |
⊢ ( ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ↔ ( 0 ≠ 0 ∨ ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ) ) |
| 55 |
52 54
|
bitr4i |
⊢ ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ↔ ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ) |
| 56 |
55
|
a1i |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ↔ ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ) ) |
| 57 |
|
opthneg |
⊢ ( ( 1 ∈ V ∧ 𝑋 ∈ 𝑊 ) → ( ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ↔ ( 1 ≠ 1 ∨ 𝑋 ≠ 𝑥 ) ) ) |
| 58 |
19 57
|
syl |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ↔ ( 1 ≠ 1 ∨ 𝑋 ≠ 𝑥 ) ) ) |
| 59 |
|
neirr |
⊢ ¬ 1 ≠ 1 |
| 60 |
59
|
biorfi |
⊢ ( 𝑋 ≠ 𝑥 ↔ ( 1 ≠ 1 ∨ 𝑋 ≠ 𝑥 ) ) |
| 61 |
58 60
|
bitr4di |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ↔ 𝑋 ≠ 𝑥 ) ) |
| 62 |
56 61
|
orbi12d |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ∨ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ) ↔ ( ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ∨ 𝑋 ≠ 𝑥 ) ) ) |
| 63 |
49 62
|
mpbird |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ∨ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ) ) |
| 64 |
22
|
olcd |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ∨ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ) ) |
| 65 |
63 64
|
jca |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ∨ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ) ∧ ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ∨ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ) ) ) |
| 66 |
|
opex |
⊢ ⟨ 1 , 𝑥 ⟩ ∈ V |
| 67 |
8 66
|
pm3.2i |
⊢ ( ⟨ 0 , 𝑥 ⟩ ∈ V ∧ ⟨ 1 , 𝑥 ⟩ ∈ V ) |
| 68 |
7 67
|
pm3.2i |
⊢ ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ∈ V ∧ ⟨ 1 , 𝑋 ⟩ ∈ V ) ∧ ( ⟨ 0 , 𝑥 ⟩ ∈ V ∧ ⟨ 1 , 𝑥 ⟩ ∈ V ) ) |
| 69 |
|
prneimg2 |
⊢ ( ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ∈ V ∧ ⟨ 1 , 𝑋 ⟩ ∈ V ) ∧ ( ⟨ 0 , 𝑥 ⟩ ∈ V ∧ ⟨ 1 , 𝑥 ⟩ ∈ V ) ) → ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ↔ ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ∨ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ) ∧ ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ∨ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ) ) ) ) |
| 70 |
68 69
|
mp1i |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ↔ ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ∨ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ) ∧ ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ∨ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 0 , 𝑥 ⟩ ) ) ) ) |
| 71 |
65 70
|
mpbird |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ) |
| 72 |
|
opex |
⊢ ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ ∈ V |
| 73 |
66 72
|
pm3.2i |
⊢ ( ⟨ 1 , 𝑥 ⟩ ∈ V ∧ ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ ∈ V ) |
| 74 |
7 73
|
pm3.2i |
⊢ ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ∈ V ∧ ⟨ 1 , 𝑋 ⟩ ∈ V ) ∧ ( ⟨ 1 , 𝑥 ⟩ ∈ V ∧ ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ ∈ V ) ) |
| 75 |
|
0ne1 |
⊢ 0 ≠ 1 |
| 76 |
75
|
orci |
⊢ ( 0 ≠ 1 ∨ ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ) |
| 77 |
50 51
|
opthne |
⊢ ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ↔ ( 0 ≠ 1 ∨ ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ 𝑥 ) ) |
| 78 |
76 77
|
mpbir |
⊢ ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , 𝑥 ⟩ |
| 79 |
75
|
orci |
⊢ ( 0 ≠ 1 ∨ ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ) |
| 80 |
50 51
|
opthne |
⊢ ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ ↔ ( 0 ≠ 1 ∨ ( ( 𝑋 + 1 ) mod 𝑁 ) ≠ ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ) ) |
| 81 |
79 80
|
mpbir |
⊢ ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ |
| 82 |
78 81
|
pm3.2i |
⊢ ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ∧ ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ ) |
| 83 |
82
|
a1i |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ∧ ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ ) ) |
| 84 |
83
|
orcd |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ∧ ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ ) ∨ ( ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ∧ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ ) ) ) |
| 85 |
|
prneimg |
⊢ ( ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ∈ V ∧ ⟨ 1 , 𝑋 ⟩ ∈ V ) ∧ ( ⟨ 1 , 𝑥 ⟩ ∈ V ∧ ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ ∈ V ) ) → ( ( ( ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ∧ ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ ≠ ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ ) ∨ ( ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 1 , 𝑥 ⟩ ∧ ⟨ 1 , 𝑋 ⟩ ≠ ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ ) ) → { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ) |
| 86 |
74 84 85
|
mpsyl |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) |
| 87 |
31 71 86
|
3jca |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∧ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∧ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ) |
| 88 |
87
|
ralrimiva |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) → ∀ 𝑥 ∈ ( 0 ..^ 𝑁 ) ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∧ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∧ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ) |
| 89 |
|
ralnex |
⊢ ( ∀ 𝑥 ∈ ( 0 ..^ 𝑁 ) ¬ ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ↔ ¬ ∃ 𝑥 ∈ ( 0 ..^ 𝑁 ) ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ) |
| 90 |
|
3ioran |
⊢ ( ¬ ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ↔ ( ¬ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∧ ¬ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∧ ¬ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ) |
| 91 |
|
df-ne |
⊢ ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ↔ ¬ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ) |
| 92 |
|
df-ne |
⊢ ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ↔ ¬ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ) |
| 93 |
|
df-ne |
⊢ ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ↔ ¬ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) |
| 94 |
91 92 93
|
3anbi123i |
⊢ ( ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∧ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∧ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ↔ ( ¬ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∧ ¬ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∧ ¬ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ) |
| 95 |
90 94
|
bitr4i |
⊢ ( ¬ ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ↔ ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∧ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∧ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ) |
| 96 |
95
|
ralbii |
⊢ ( ∀ 𝑥 ∈ ( 0 ..^ 𝑁 ) ¬ ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ↔ ∀ 𝑥 ∈ ( 0 ..^ 𝑁 ) ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∧ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∧ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ) |
| 97 |
89 96
|
bitr3i |
⊢ ( ¬ ∃ 𝑥 ∈ ( 0 ..^ 𝑁 ) ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ↔ ∀ 𝑥 ∈ ( 0 ..^ 𝑁 ) ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∧ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∧ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ≠ { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ) |
| 98 |
88 97
|
sylibr |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) → ¬ ∃ 𝑥 ∈ ( 0 ..^ 𝑁 ) ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ) |
| 99 |
|
eqid |
⊢ ( 0 ..^ 𝑁 ) = ( 0 ..^ 𝑁 ) |
| 100 |
99 1 2 4
|
gpgedgel |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ) → ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ∈ 𝐸 ↔ ∃ 𝑥 ∈ ( 0 ..^ 𝑁 ) ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ) ) |
| 101 |
100
|
3adant3 |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) → ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ∈ 𝐸 ↔ ∃ 𝑥 ∈ ( 0 ..^ 𝑁 ) ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 0 , ( ( 𝑥 + 1 ) mod 𝑁 ) ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 0 , 𝑥 ⟩ , ⟨ 1 , 𝑥 ⟩ } ∨ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } = { ⟨ 1 , 𝑥 ⟩ , ⟨ 1 , ( ( 𝑥 + 𝐾 ) mod 𝑁 ) ⟩ } ) ) ) |
| 102 |
98 101
|
mtbird |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) → ¬ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ∈ 𝐸 ) |
| 103 |
|
df-nel |
⊢ ( { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ∉ 𝐸 ↔ ¬ { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ∈ 𝐸 ) |
| 104 |
102 103
|
sylibr |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊 ) → { ⟨ 0 , ( ( 𝑋 + 1 ) mod 𝑁 ) ⟩ , ⟨ 1 , 𝑋 ⟩ } ∉ 𝐸 ) |