Step |
Hyp |
Ref |
Expression |
1 |
|
vdwnn.1 |
⊢ ( 𝜑 → 𝑅 ∈ Fin ) |
2 |
|
vdwnn.2 |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ 𝑅 ) |
3 |
|
vdwnn.3 |
⊢ 𝑆 = { 𝑘 ∈ ℕ ∣ ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝑘 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) } |
4 |
|
eluzel2 |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → 𝐴 ∈ ℤ ) |
5 |
|
peano2zm |
⊢ ( 𝐴 ∈ ℤ → ( 𝐴 − 1 ) ∈ ℤ ) |
6 |
4 5
|
syl |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐴 − 1 ) ∈ ℤ ) |
7 |
|
id |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) |
8 |
4
|
zcnd |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → 𝐴 ∈ ℂ ) |
9 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
10 |
|
npcan |
⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝐴 − 1 ) + 1 ) = 𝐴 ) |
11 |
8 9 10
|
sylancl |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( ( 𝐴 − 1 ) + 1 ) = 𝐴 ) |
12 |
11
|
fveq2d |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( ℤ≥ ‘ ( ( 𝐴 − 1 ) + 1 ) ) = ( ℤ≥ ‘ 𝐴 ) ) |
13 |
7 12
|
eleqtrrd |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → 𝐵 ∈ ( ℤ≥ ‘ ( ( 𝐴 − 1 ) + 1 ) ) ) |
14 |
|
eluzp1m1 |
⊢ ( ( ( 𝐴 − 1 ) ∈ ℤ ∧ 𝐵 ∈ ( ℤ≥ ‘ ( ( 𝐴 − 1 ) + 1 ) ) ) → ( 𝐵 − 1 ) ∈ ( ℤ≥ ‘ ( 𝐴 − 1 ) ) ) |
15 |
6 13 14
|
syl2anc |
⊢ ( 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) → ( 𝐵 − 1 ) ∈ ( ℤ≥ ‘ ( 𝐴 − 1 ) ) ) |
16 |
15
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) ∧ 𝐴 ∈ ℕ ) → ( 𝐵 − 1 ) ∈ ( ℤ≥ ‘ ( 𝐴 − 1 ) ) ) |
17 |
|
fzss2 |
⊢ ( ( 𝐵 − 1 ) ∈ ( ℤ≥ ‘ ( 𝐴 − 1 ) ) → ( 0 ... ( 𝐴 − 1 ) ) ⊆ ( 0 ... ( 𝐵 − 1 ) ) ) |
18 |
|
ssralv |
⊢ ( ( 0 ... ( 𝐴 − 1 ) ) ⊆ ( 0 ... ( 𝐵 − 1 ) ) → ( ∀ 𝑚 ∈ ( 0 ... ( 𝐵 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) → ∀ 𝑚 ∈ ( 0 ... ( 𝐴 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
19 |
16 17 18
|
3syl |
⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) ∧ 𝐴 ∈ ℕ ) → ( ∀ 𝑚 ∈ ( 0 ... ( 𝐵 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) → ∀ 𝑚 ∈ ( 0 ... ( 𝐴 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
20 |
19
|
reximdv |
⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) ∧ 𝐴 ∈ ℕ ) → ( ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐵 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) → ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐴 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
21 |
20
|
reximdv |
⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) ∧ 𝐴 ∈ ℕ ) → ( ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐵 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) → ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐴 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
22 |
21
|
con3d |
⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) ∧ 𝐴 ∈ ℕ ) → ( ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐴 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) → ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐵 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
23 |
|
id |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℕ ) |
24 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) → 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) |
25 |
|
eluznn |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) → 𝐵 ∈ ℕ ) |
26 |
23 24 25
|
syl2anr |
⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) ∧ 𝐴 ∈ ℕ ) → 𝐵 ∈ ℕ ) |
27 |
22 26
|
jctild |
⊢ ( ( ( 𝜑 ∧ 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) ∧ 𝐴 ∈ ℕ ) → ( ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐴 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) → ( 𝐵 ∈ ℕ ∧ ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐵 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) ) |
28 |
27
|
expimpd |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) → ( ( 𝐴 ∈ ℕ ∧ ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐴 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) → ( 𝐵 ∈ ℕ ∧ ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐵 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) ) |
29 |
|
oveq1 |
⊢ ( 𝑘 = 𝐴 → ( 𝑘 − 1 ) = ( 𝐴 − 1 ) ) |
30 |
29
|
oveq2d |
⊢ ( 𝑘 = 𝐴 → ( 0 ... ( 𝑘 − 1 ) ) = ( 0 ... ( 𝐴 − 1 ) ) ) |
31 |
30
|
raleqdv |
⊢ ( 𝑘 = 𝐴 → ( ∀ 𝑚 ∈ ( 0 ... ( 𝑘 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ↔ ∀ 𝑚 ∈ ( 0 ... ( 𝐴 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
32 |
31
|
2rexbidv |
⊢ ( 𝑘 = 𝐴 → ( ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝑘 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ↔ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐴 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
33 |
32
|
notbid |
⊢ ( 𝑘 = 𝐴 → ( ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝑘 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ↔ ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐴 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
34 |
33 3
|
elrab2 |
⊢ ( 𝐴 ∈ 𝑆 ↔ ( 𝐴 ∈ ℕ ∧ ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐴 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
35 |
|
oveq1 |
⊢ ( 𝑘 = 𝐵 → ( 𝑘 − 1 ) = ( 𝐵 − 1 ) ) |
36 |
35
|
oveq2d |
⊢ ( 𝑘 = 𝐵 → ( 0 ... ( 𝑘 − 1 ) ) = ( 0 ... ( 𝐵 − 1 ) ) ) |
37 |
36
|
raleqdv |
⊢ ( 𝑘 = 𝐵 → ( ∀ 𝑚 ∈ ( 0 ... ( 𝑘 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ↔ ∀ 𝑚 ∈ ( 0 ... ( 𝐵 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
38 |
37
|
2rexbidv |
⊢ ( 𝑘 = 𝐵 → ( ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝑘 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ↔ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐵 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
39 |
38
|
notbid |
⊢ ( 𝑘 = 𝐵 → ( ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝑘 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ↔ ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐵 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
40 |
39 3
|
elrab2 |
⊢ ( 𝐵 ∈ 𝑆 ↔ ( 𝐵 ∈ ℕ ∧ ¬ ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐵 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝐹 “ { 𝑐 } ) ) ) |
41 |
28 34 40
|
3imtr4g |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( ℤ≥ ‘ 𝐴 ) ) → ( 𝐴 ∈ 𝑆 → 𝐵 ∈ 𝑆 ) ) |