Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) → 𝑅 ∈ Fin ) |
2 |
|
simpr |
⊢ ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) → 𝐾 ∈ ℕ0 ) |
3 |
1 2
|
vdwlem13 |
⊢ ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) → ∃ 𝑛 ∈ ℕ ∀ 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) 𝐾 MonoAP 𝑓 ) |
4 |
|
ovex |
⊢ ( 1 ... 𝑛 ) ∈ V |
5 |
|
simpllr |
⊢ ( ( ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) ) → 𝐾 ∈ ℕ0 ) |
6 |
|
simpll |
⊢ ( ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ ) → 𝑅 ∈ Fin ) |
7 |
|
elmapg |
⊢ ( ( 𝑅 ∈ Fin ∧ ( 1 ... 𝑛 ) ∈ V ) → ( 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) ↔ 𝑓 : ( 1 ... 𝑛 ) ⟶ 𝑅 ) ) |
8 |
6 4 7
|
sylancl |
⊢ ( ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ ) → ( 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) ↔ 𝑓 : ( 1 ... 𝑛 ) ⟶ 𝑅 ) ) |
9 |
8
|
biimpa |
⊢ ( ( ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) ) → 𝑓 : ( 1 ... 𝑛 ) ⟶ 𝑅 ) |
10 |
|
simplr |
⊢ ( ( ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) ) → 𝑛 ∈ ℕ ) |
11 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
12 |
10 11
|
eleqtrdi |
⊢ ( ( ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) |
13 |
|
eluzfz1 |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... 𝑛 ) ) |
14 |
12 13
|
syl |
⊢ ( ( ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) ) → 1 ∈ ( 1 ... 𝑛 ) ) |
15 |
4 5 9 14
|
vdwmc2 |
⊢ ( ( ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) ) → ( 𝐾 MonoAP 𝑓 ↔ ∃ 𝑐 ∈ 𝑅 ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝑓 “ { 𝑐 } ) ) ) |
16 |
15
|
ralbidva |
⊢ ( ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ ) → ( ∀ 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) 𝐾 MonoAP 𝑓 ↔ ∀ 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) ∃ 𝑐 ∈ 𝑅 ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝑓 “ { 𝑐 } ) ) ) |
17 |
16
|
rexbidva |
⊢ ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) → ( ∃ 𝑛 ∈ ℕ ∀ 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) 𝐾 MonoAP 𝑓 ↔ ∃ 𝑛 ∈ ℕ ∀ 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) ∃ 𝑐 ∈ 𝑅 ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝑓 “ { 𝑐 } ) ) ) |
18 |
3 17
|
mpbid |
⊢ ( ( 𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0 ) → ∃ 𝑛 ∈ ℕ ∀ 𝑓 ∈ ( 𝑅 ↑m ( 1 ... 𝑛 ) ) ∃ 𝑐 ∈ 𝑅 ∃ 𝑎 ∈ ℕ ∃ 𝑑 ∈ ℕ ∀ 𝑚 ∈ ( 0 ... ( 𝐾 − 1 ) ) ( 𝑎 + ( 𝑚 · 𝑑 ) ) ∈ ( ◡ 𝑓 “ { 𝑐 } ) ) |