Step |
Hyp |
Ref |
Expression |
1 |
|
clwlkclwwlklem2.f |
⊢ 𝐹 = ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ↦ if ( 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) , ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 + 1 ) ) } ) , ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ 0 ) } ) ) ) |
2 |
|
simpr |
⊢ ( ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) ) → 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) ) |
3 |
|
nn0z |
⊢ ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 → ( ♯ ‘ 𝑃 ) ∈ ℤ ) |
4 |
|
2z |
⊢ 2 ∈ ℤ |
5 |
3 4
|
jctir |
⊢ ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 → ( ( ♯ ‘ 𝑃 ) ∈ ℤ ∧ 2 ∈ ℤ ) ) |
6 |
|
zsubcl |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℤ ∧ 2 ∈ ℤ ) → ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ℤ ) |
7 |
5 6
|
syl |
⊢ ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 → ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ℤ ) |
8 |
7
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ℤ ) |
9 |
8
|
adantr |
⊢ ( ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) ) → ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ℤ ) |
10 |
2 9
|
eqeltrd |
⊢ ( ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) ) → 𝑥 ∈ ℤ ) |
11 |
10
|
ex |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) → 𝑥 ∈ ℤ ) ) |
12 |
|
zre |
⊢ ( 𝑥 ∈ ℤ → 𝑥 ∈ ℝ ) |
13 |
|
nn0re |
⊢ ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 → ( ♯ ‘ 𝑃 ) ∈ ℝ ) |
14 |
|
2re |
⊢ 2 ∈ ℝ |
15 |
14
|
a1i |
⊢ ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 → 2 ∈ ℝ ) |
16 |
13 15
|
resubcld |
⊢ ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 → ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ℝ ) |
17 |
16
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ℝ ) |
18 |
|
lttri3 |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ℝ ) → ( 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) ↔ ( ¬ 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) ∧ ¬ ( ( ♯ ‘ 𝑃 ) − 2 ) < 𝑥 ) ) ) |
19 |
12 17 18
|
syl2anr |
⊢ ( ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑥 ∈ ℤ ) → ( 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) ↔ ( ¬ 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) ∧ ¬ ( ( ♯ ‘ 𝑃 ) − 2 ) < 𝑥 ) ) ) |
20 |
|
simpl |
⊢ ( ( ¬ 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) ∧ ¬ ( ( ♯ ‘ 𝑃 ) − 2 ) < 𝑥 ) → ¬ 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) ) |
21 |
19 20
|
syl6bi |
⊢ ( ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑥 ∈ ℤ ) → ( 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) → ¬ 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) ) ) |
22 |
21
|
ex |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( 𝑥 ∈ ℤ → ( 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) → ¬ 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) ) ) ) |
23 |
11 22
|
syld |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) → ( 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) → ¬ 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) ) ) ) |
24 |
23
|
com13 |
⊢ ( 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) → ( 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) → ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ¬ 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) ) ) ) |
25 |
24
|
pm2.43i |
⊢ ( 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) → ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ¬ 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) ) ) |
26 |
25
|
impcom |
⊢ ( ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) ) → ¬ 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) ) |
27 |
26
|
iffalsed |
⊢ ( ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) ) → if ( 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) , ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 + 1 ) ) } ) , ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ 0 ) } ) ) = ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ 0 ) } ) ) |
28 |
|
fveq2 |
⊢ ( 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) → ( 𝑃 ‘ 𝑥 ) = ( 𝑃 ‘ ( ( ♯ ‘ 𝑃 ) − 2 ) ) ) |
29 |
28
|
adantl |
⊢ ( ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) ) → ( 𝑃 ‘ 𝑥 ) = ( 𝑃 ‘ ( ( ♯ ‘ 𝑃 ) − 2 ) ) ) |
30 |
29
|
preq1d |
⊢ ( ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) ) → { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ 0 ) } = { ( 𝑃 ‘ ( ( ♯ ‘ 𝑃 ) − 2 ) ) , ( 𝑃 ‘ 0 ) } ) |
31 |
30
|
fveq2d |
⊢ ( ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) ) → ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ 0 ) } ) = ( ◡ 𝐸 ‘ { ( 𝑃 ‘ ( ( ♯ ‘ 𝑃 ) − 2 ) ) , ( 𝑃 ‘ 0 ) } ) ) |
32 |
27 31
|
eqtrd |
⊢ ( ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑥 = ( ( ♯ ‘ 𝑃 ) − 2 ) ) → if ( 𝑥 < ( ( ♯ ‘ 𝑃 ) − 2 ) , ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 + 1 ) ) } ) , ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ 0 ) } ) ) = ( ◡ 𝐸 ‘ { ( 𝑃 ‘ ( ( ♯ ‘ 𝑃 ) − 2 ) ) , ( 𝑃 ‘ 0 ) } ) ) |
33 |
5
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( ( ♯ ‘ 𝑃 ) ∈ ℤ ∧ 2 ∈ ℤ ) ) |
34 |
33 6
|
syl |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ℤ ) |
35 |
13 15
|
subge0d |
⊢ ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 → ( 0 ≤ ( ( ♯ ‘ 𝑃 ) − 2 ) ↔ 2 ≤ ( ♯ ‘ 𝑃 ) ) ) |
36 |
35
|
biimpar |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → 0 ≤ ( ( ♯ ‘ 𝑃 ) − 2 ) ) |
37 |
|
elnn0z |
⊢ ( ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ℕ0 ↔ ( ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ℤ ∧ 0 ≤ ( ( ♯ ‘ 𝑃 ) − 2 ) ) ) |
38 |
34 36 37
|
sylanbrc |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ℕ0 ) |
39 |
|
nn0ge2m1nn |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( ( ♯ ‘ 𝑃 ) − 1 ) ∈ ℕ ) |
40 |
|
1red |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → 1 ∈ ℝ ) |
41 |
14
|
a1i |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → 2 ∈ ℝ ) |
42 |
13
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( ♯ ‘ 𝑃 ) ∈ ℝ ) |
43 |
|
1lt2 |
⊢ 1 < 2 |
44 |
43
|
a1i |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → 1 < 2 ) |
45 |
40 41 42 44
|
ltsub2dd |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( ( ♯ ‘ 𝑃 ) − 2 ) < ( ( ♯ ‘ 𝑃 ) − 1 ) ) |
46 |
|
elfzo0 |
⊢ ( ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ↔ ( ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ℕ0 ∧ ( ( ♯ ‘ 𝑃 ) − 1 ) ∈ ℕ ∧ ( ( ♯ ‘ 𝑃 ) − 2 ) < ( ( ♯ ‘ 𝑃 ) − 1 ) ) ) |
47 |
38 39 45 46
|
syl3anbrc |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( ( ♯ ‘ 𝑃 ) − 2 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ) |
48 |
|
fvexd |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( ◡ 𝐸 ‘ { ( 𝑃 ‘ ( ( ♯ ‘ 𝑃 ) − 2 ) ) , ( 𝑃 ‘ 0 ) } ) ∈ V ) |
49 |
1 32 47 48
|
fvmptd2 |
⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( 𝐹 ‘ ( ( ♯ ‘ 𝑃 ) − 2 ) ) = ( ◡ 𝐸 ‘ { ( 𝑃 ‘ ( ( ♯ ‘ 𝑃 ) − 2 ) ) , ( 𝑃 ‘ 0 ) } ) ) |