| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | vdw.r | . . 3
⊢ (𝜑 → 𝑅 ∈ Fin) | 
| 2 |  | vdwlem9.k | . . 3
⊢ (𝜑 → 𝐾 ∈
(ℤ≥‘2)) | 
| 3 |  | vdwlem9.s | . . 3
⊢ (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑m (1...𝑛))𝐾 MonoAP 𝑓) | 
| 4 |  | hashcl 14395 | . . . . 5
⊢ (𝑅 ∈ Fin →
(♯‘𝑅) ∈
ℕ0) | 
| 5 | 1, 4 | syl 17 | . . . 4
⊢ (𝜑 → (♯‘𝑅) ∈
ℕ0) | 
| 6 |  | nn0p1nn 12565 | . . . 4
⊢
((♯‘𝑅)
∈ ℕ0 → ((♯‘𝑅) + 1) ∈ ℕ) | 
| 7 | 5, 6 | syl 17 | . . 3
⊢ (𝜑 → ((♯‘𝑅) + 1) ∈
ℕ) | 
| 8 | 1, 2, 3, 7 | vdwlem10 17028 | . 2
⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))(〈((♯‘𝑅) + 1), 𝐾〉 PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) | 
| 9 | 1 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ Fin) | 
| 10 |  | ovex 7464 | . . . . . . 7
⊢
(1...𝑛) ∈
V | 
| 11 |  | elmapg 8879 | . . . . . . 7
⊢ ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → (𝑓 ∈ (𝑅 ↑m (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅)) | 
| 12 | 9, 10, 11 | sylancl 586 | . . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑓 ∈ (𝑅 ↑m (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅)) | 
| 13 | 12 | biimpa 476 | . . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑m (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅) | 
| 14 | 5 | nn0red 12588 | . . . . . . . . . . 11
⊢ (𝜑 → (♯‘𝑅) ∈
ℝ) | 
| 15 | 14 | ltp1d 12198 | . . . . . . . . . 10
⊢ (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1)) | 
| 16 |  | peano2re 11434 | . . . . . . . . . . . 12
⊢
((♯‘𝑅)
∈ ℝ → ((♯‘𝑅) + 1) ∈ ℝ) | 
| 17 | 14, 16 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → ((♯‘𝑅) + 1) ∈
ℝ) | 
| 18 | 14, 17 | ltnled 11408 | . . . . . . . . . 10
⊢ (𝜑 → ((♯‘𝑅) < ((♯‘𝑅) + 1) ↔ ¬
((♯‘𝑅) + 1)
≤ (♯‘𝑅))) | 
| 19 | 15, 18 | mpbid 232 | . . . . . . . . 9
⊢ (𝜑 → ¬
((♯‘𝑅) + 1)
≤ (♯‘𝑅)) | 
| 20 | 19 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅)) | 
| 21 |  | eluz2nn 12924 | . . . . . . . . . . . . 13
⊢ (𝐾 ∈
(ℤ≥‘2) → 𝐾 ∈ ℕ) | 
| 22 | 2, 21 | syl 17 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐾 ∈ ℕ) | 
| 23 | 22 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ) | 
| 24 | 23 | nnnn0d 12587 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈
ℕ0) | 
| 25 |  | simprr 773 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝑓:(1...𝑛)⟶𝑅) | 
| 26 | 7 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((♯‘𝑅) + 1) ∈ ℕ) | 
| 27 |  | eqid 2737 | . . . . . . . . . 10
⊢
(1...((♯‘𝑅) + 1)) = (1...((♯‘𝑅) + 1)) | 
| 28 | 10, 24, 25, 26, 27 | vdwpc 17018 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (〈((♯‘𝑅) + 1), 𝐾〉 PolyAP 𝑓 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1)))(∀𝑖 ∈
(1...((♯‘𝑅) +
1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈
(1...((♯‘𝑅) +
1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = ((♯‘𝑅) + 1)))) | 
| 29 | 1 | ad3antrrr 730 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ ∀𝑖
∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → 𝑅 ∈ Fin) | 
| 30 | 25 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) → 𝑓:(1...𝑛)⟶𝑅) | 
| 31 | 25 | ad3antrrr 730 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → 𝑓:(1...𝑛)⟶𝑅) | 
| 32 |  | simpr 484 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) | 
| 33 |  | cnvimass 6100 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ⊆ dom 𝑓 | 
| 34 | 32, 33 | sstrdi 3996 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ dom 𝑓) | 
| 35 | 31, 34 | fssdmd 6754 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (1...𝑛)) | 
| 36 | 22 | ad3antrrr 730 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) → 𝐾 ∈
ℕ) | 
| 37 |  | simplrl 777 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) → 𝑎 ∈
ℕ) | 
| 38 |  | simprr 773 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) → 𝑑 ∈
(ℕ ↑m (1...((♯‘𝑅) + 1)))) | 
| 39 |  | nnex 12272 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ℕ
∈ V | 
| 40 |  | ovex 7464 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(1...((♯‘𝑅) + 1)) ∈ V | 
| 41 | 39, 40 | elmap 8911 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑑 ∈ (ℕ
↑m (1...((♯‘𝑅) + 1))) ↔ 𝑑:(1...((♯‘𝑅) + 1))⟶ℕ) | 
| 42 | 38, 41 | sylib 218 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) → 𝑑:(1...((♯‘𝑅) + 1))⟶ℕ) | 
| 43 | 42 | ffvelcdmda 7104 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) → (𝑑‘𝑖) ∈
ℕ) | 
| 44 | 37, 43 | nnaddcld 12318 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) → (𝑎 + (𝑑‘𝑖)) ∈ ℕ) | 
| 45 |  | vdwapid1 17013 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐾 ∈ ℕ ∧ (𝑎 + (𝑑‘𝑖)) ∈ ℕ ∧ (𝑑‘𝑖) ∈ ℕ) → (𝑎 + (𝑑‘𝑖)) ∈ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖))) | 
| 46 | 36, 44, 43, 45 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) → (𝑎 + (𝑑‘𝑖)) ∈ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖))) | 
| 47 | 46 | adantr 480 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → (𝑎 + (𝑑‘𝑖)) ∈ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖))) | 
| 48 | 35, 47 | sseldd 3984 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → (𝑎 + (𝑑‘𝑖)) ∈ (1...𝑛)) | 
| 49 | 48 | ex 412 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) → (((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) → (𝑎 + (𝑑‘𝑖)) ∈ (1...𝑛))) | 
| 50 |  | ffvelcdm 7101 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓:(1...𝑛)⟶𝑅 ∧ (𝑎 + (𝑑‘𝑖)) ∈ (1...𝑛)) → (𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅) | 
| 51 | 30, 49, 50 | syl6an 684 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ 𝑖 ∈
(1...((♯‘𝑅) +
1))) → (((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) → (𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅)) | 
| 52 | 51 | ralimdva 3167 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) → (∀𝑖
∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) → ∀𝑖 ∈ (1...((♯‘𝑅) + 1))(𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅)) | 
| 53 | 52 | imp 406 | . . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ ∀𝑖
∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ∀𝑖 ∈ (1...((♯‘𝑅) + 1))(𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅) | 
| 54 |  | eqid 2737 | . . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈
(1...((♯‘𝑅) +
1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) = (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) | 
| 55 | 54 | fmpt 7130 | . . . . . . . . . . . . . . . 16
⊢
(∀𝑖 ∈
(1...((♯‘𝑅) +
1))(𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅 ↔ (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))):(1...((♯‘𝑅) + 1))⟶𝑅) | 
| 56 | 53, 55 | sylib 218 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ ∀𝑖
∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))):(1...((♯‘𝑅) + 1))⟶𝑅) | 
| 57 | 56 | frnd 6744 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ ∀𝑖
∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ⊆ 𝑅) | 
| 58 |  | ssdomg 9040 | . . . . . . . . . . . . . 14
⊢ (𝑅 ∈ Fin → (ran (𝑖 ∈
(1...((♯‘𝑅) +
1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ⊆ 𝑅 → ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ≼ 𝑅)) | 
| 59 | 29, 57, 58 | sylc 65 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ ∀𝑖
∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ≼ 𝑅) | 
| 60 | 29, 57 | ssfid 9301 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ ∀𝑖
∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ∈ Fin) | 
| 61 |  | hashdom 14418 | . . . . . . . . . . . . . 14
⊢ ((ran
(𝑖 ∈
(1...((♯‘𝑅) +
1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ∈ Fin ∧ 𝑅 ∈ Fin) → ((♯‘ran
(𝑖 ∈
(1...((♯‘𝑅) +
1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) ≤ (♯‘𝑅) ↔ ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ≼ 𝑅)) | 
| 62 | 60, 29, 61 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ ∀𝑖
∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ((♯‘ran (𝑖 ∈
(1...((♯‘𝑅) +
1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) ≤ (♯‘𝑅) ↔ ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ≼ 𝑅)) | 
| 63 | 59, 62 | mpbird 257 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ ∀𝑖
∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → (♯‘ran (𝑖 ∈
(1...((♯‘𝑅) +
1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) ≤ (♯‘𝑅)) | 
| 64 |  | breq1 5146 | . . . . . . . . . . . 12
⊢
((♯‘ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = ((♯‘𝑅) + 1) → ((♯‘ran (𝑖 ∈
(1...((♯‘𝑅) +
1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) ≤ (♯‘𝑅) ↔ ((♯‘𝑅) + 1) ≤ (♯‘𝑅))) | 
| 65 | 63, 64 | syl5ibcom 245 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) ∧ ∀𝑖
∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ((♯‘ran (𝑖 ∈
(1...((♯‘𝑅) +
1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = ((♯‘𝑅) + 1) → ((♯‘𝑅) + 1) ≤ (♯‘𝑅))) | 
| 66 | 65 | expimpd 453 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1))))) → ((∀𝑖
∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈
(1...((♯‘𝑅) +
1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = ((♯‘𝑅) + 1)) → ((♯‘𝑅) + 1) ≤ (♯‘𝑅))) | 
| 67 | 66 | rexlimdvva 3213 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m
(1...((♯‘𝑅) +
1)))(∀𝑖 ∈
(1...((♯‘𝑅) +
1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈
(1...((♯‘𝑅) +
1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = ((♯‘𝑅) + 1)) → ((♯‘𝑅) + 1) ≤ (♯‘𝑅))) | 
| 68 | 28, 67 | sylbid 240 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (〈((♯‘𝑅) + 1), 𝐾〉 PolyAP 𝑓 → ((♯‘𝑅) + 1) ≤ (♯‘𝑅))) | 
| 69 | 20, 68 | mtod 198 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ 〈((♯‘𝑅) + 1), 𝐾〉 PolyAP 𝑓) | 
| 70 |  | biorf 937 | . . . . . . 7
⊢ (¬
〈((♯‘𝑅) +
1), 𝐾〉 PolyAP 𝑓 → ((𝐾 + 1) MonoAP 𝑓 ↔ (〈((♯‘𝑅) + 1), 𝐾〉 PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))) | 
| 71 | 69, 70 | syl 17 | . . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((𝐾 + 1) MonoAP 𝑓 ↔ (〈((♯‘𝑅) + 1), 𝐾〉 PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))) | 
| 72 | 71 | anassrs 467 | . . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓:(1...𝑛)⟶𝑅) → ((𝐾 + 1) MonoAP 𝑓 ↔ (〈((♯‘𝑅) + 1), 𝐾〉 PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))) | 
| 73 | 13, 72 | syldan 591 | . . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑m (1...𝑛))) → ((𝐾 + 1) MonoAP 𝑓 ↔ (〈((♯‘𝑅) + 1), 𝐾〉 PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))) | 
| 74 | 73 | ralbidva 3176 | . . 3
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∀𝑓 ∈ (𝑅 ↑m (1...𝑛))(𝐾 + 1) MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))(〈((♯‘𝑅) + 1), 𝐾〉 PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))) | 
| 75 | 74 | rexbidva 3177 | . 2
⊢ (𝜑 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))(𝐾 + 1) MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))(〈((♯‘𝑅) + 1), 𝐾〉 PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))) | 
| 76 | 8, 75 | mpbird 257 | 1
⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑m (1...𝑛))(𝐾 + 1) MonoAP 𝑓) |