MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vdwlem11 Structured version   Visualization version   GIF version

Theorem vdwlem11 15988
Description: Lemma for vdw 15991. (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
vdw.r (𝜑𝑅 ∈ Fin)
vdwlem9.k (𝜑𝐾 ∈ (ℤ‘2))
vdwlem9.s (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
Assertion
Ref Expression
vdwlem11 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(𝐾 + 1) MonoAP 𝑓)
Distinct variable groups:   𝜑,𝑛,𝑓   𝑓,𝑠,𝐾,𝑛   𝑅,𝑓,𝑛,𝑠   𝜑,𝑓
Allowed substitution hint:   𝜑(𝑠)

Proof of Theorem vdwlem11
Dummy variables 𝑎 𝑑 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdw.r . . 3 (𝜑𝑅 ∈ Fin)
2 vdwlem9.k . . 3 (𝜑𝐾 ∈ (ℤ‘2))
3 vdwlem9.s . . 3 (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
4 hashcl 13354 . . . . 5 (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ0)
51, 4syl 17 . . . 4 (𝜑 → (♯‘𝑅) ∈ ℕ0)
6 nn0p1nn 11583 . . . 4 ((♯‘𝑅) ∈ ℕ0 → ((♯‘𝑅) + 1) ∈ ℕ)
75, 6syl 17 . . 3 (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
81, 2, 3, 7vdwlem10 15987 . 2 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
91adantr 472 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑅 ∈ Fin)
10 ovex 6878 . . . . . . 7 (1...𝑛) ∈ V
11 elmapg 8077 . . . . . . 7 ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → (𝑓 ∈ (𝑅𝑚 (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
129, 10, 11sylancl 580 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑓 ∈ (𝑅𝑚 (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
1312biimpa 468 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅𝑚 (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅)
145nn0red 11603 . . . . . . . . . . 11 (𝜑 → (♯‘𝑅) ∈ ℝ)
1514ltp1d 11212 . . . . . . . . . 10 (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
16 peano2re 10467 . . . . . . . . . . . 12 ((♯‘𝑅) ∈ ℝ → ((♯‘𝑅) + 1) ∈ ℝ)
1714, 16syl 17 . . . . . . . . . . 11 (𝜑 → ((♯‘𝑅) + 1) ∈ ℝ)
1814, 17ltnled 10442 . . . . . . . . . 10 (𝜑 → ((♯‘𝑅) < ((♯‘𝑅) + 1) ↔ ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
1915, 18mpbid 223 . . . . . . . . 9 (𝜑 → ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅))
2019adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅))
21 eluz2nn 11931 . . . . . . . . . . . . 13 (𝐾 ∈ (ℤ‘2) → 𝐾 ∈ ℕ)
222, 21syl 17 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℕ)
2322adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ)
2423nnnn0d 11602 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ0)
25 simprr 789 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝑓:(1...𝑛)⟶𝑅)
267adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((♯‘𝑅) + 1) ∈ ℕ)
27 eqid 2765 . . . . . . . . . 10 (1...((♯‘𝑅) + 1)) = (1...((♯‘𝑅) + 1))
2810, 24, 25, 26, 27vdwpc 15977 . . . . . . . . 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 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = ((♯‘𝑅) + 1))))
291ad3antrrr 721 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → 𝑅 ∈ Fin)
3025ad2antrr 717 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → 𝑓:(1...𝑛)⟶𝑅)
3125ad3antrrr 721 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) ∧ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → 𝑓:(1...𝑛)⟶𝑅)
32 simpr 477 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) ∧ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}))
33 cnvimass 5669 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ⊆ dom 𝑓
3432, 33syl6ss 3775 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) ∧ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ dom 𝑓)
3531, 34fssdmd 6240 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) ∧ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (1...𝑛))
3622ad3antrrr 721 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → 𝐾 ∈ ℕ)
37 simplrl 795 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → 𝑎 ∈ ℕ)
38 simprr 789 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) → 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))
39 nnex 11285 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ℕ ∈ V
40 ovex 6878 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (1...((♯‘𝑅) + 1)) ∈ V
4139, 40elmap 8093 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))) ↔ 𝑑:(1...((♯‘𝑅) + 1))⟶ℕ)
4238, 41sylib 209 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) → 𝑑:(1...((♯‘𝑅) + 1))⟶ℕ)
4342ffvelrnda 6553 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → (𝑑𝑖) ∈ ℕ)
4437, 43nnaddcld 11328 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → (𝑎 + (𝑑𝑖)) ∈ ℕ)
45 vdwapid1 15972 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐾 ∈ ℕ ∧ (𝑎 + (𝑑𝑖)) ∈ ℕ ∧ (𝑑𝑖) ∈ ℕ) → (𝑎 + (𝑑𝑖)) ∈ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
4636, 44, 43, 45syl3anc 1490 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → (𝑎 + (𝑑𝑖)) ∈ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
4746adantr 472 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) ∧ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → (𝑎 + (𝑑𝑖)) ∈ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
4835, 47sseldd 3764 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) ∧ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → (𝑎 + (𝑑𝑖)) ∈ (1...𝑛))
4948ex 401 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → (((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) → (𝑎 + (𝑑𝑖)) ∈ (1...𝑛)))
50 ffvelrn 6551 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑛)⟶𝑅 ∧ (𝑎 + (𝑑𝑖)) ∈ (1...𝑛)) → (𝑓‘(𝑎 + (𝑑𝑖))) ∈ 𝑅)
5130, 49, 50syl6an 674 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → (((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) → (𝑓‘(𝑎 + (𝑑𝑖))) ∈ 𝑅))
5251ralimdva 3109 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) → (∀𝑖 ∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) → ∀𝑖 ∈ (1...((♯‘𝑅) + 1))(𝑓‘(𝑎 + (𝑑𝑖))) ∈ 𝑅))
5352imp 395 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → ∀𝑖 ∈ (1...((♯‘𝑅) + 1))(𝑓‘(𝑎 + (𝑑𝑖))) ∈ 𝑅)
54 eqid 2765 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))
5554fmpt 6574 . . . . . . . . . . . . . . . 16 (∀𝑖 ∈ (1...((♯‘𝑅) + 1))(𝑓‘(𝑎 + (𝑑𝑖))) ∈ 𝑅 ↔ (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))):(1...((♯‘𝑅) + 1))⟶𝑅)
5653, 55sylib 209 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))):(1...((♯‘𝑅) + 1))⟶𝑅)
5756frnd 6232 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ⊆ 𝑅)
58 ssdomg 8210 . . . . . . . . . . . . . 14 (𝑅 ∈ Fin → (ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ⊆ 𝑅 → ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ≼ 𝑅))
5929, 57, 58sylc 65 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ≼ 𝑅)
60 ssfi 8391 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Fin ∧ ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ⊆ 𝑅) → ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ∈ Fin)
6129, 57, 60syl2anc 579 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ∈ Fin)
62 hashdom 13375 . . . . . . . . . . . . . 14 ((ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ∈ Fin ∧ 𝑅 ∈ Fin) → ((♯‘ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) ≤ (♯‘𝑅) ↔ ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ≼ 𝑅))
6361, 29, 62syl2anc 579 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → ((♯‘ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) ≤ (♯‘𝑅) ↔ ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ≼ 𝑅))
6459, 63mpbird 248 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → (♯‘ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) ≤ (♯‘𝑅))
65 breq1 4814 . . . . . . . . . . . 12 ((♯‘ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = ((♯‘𝑅) + 1) → ((♯‘ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) ≤ (♯‘𝑅) ↔ ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
6664, 65syl5ibcom 236 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))})) → ((♯‘ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = ((♯‘𝑅) + 1) → ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
6766expimpd 445 . . . . . . . . . 10 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1))))) → ((∀𝑖 ∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = ((♯‘𝑅) + 1)) → ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
6867rexlimdvva 3185 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚 (1...((♯‘𝑅) + 1)))(∀𝑖 ∈ (1...((♯‘𝑅) + 1))((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = ((♯‘𝑅) + 1)) → ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
6928, 68sylbid 231 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
7020, 69mtod 189 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓)
71 biorf 960 . . . . . . 7 (¬ ⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7270, 71syl 17 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7372anassrs 459 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓:(1...𝑛)⟶𝑅) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7413, 73syldan 585 . . . 4 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅𝑚 (1...𝑛))) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7574ralbidva 3132 . . 3 ((𝜑𝑛 ∈ ℕ) → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(𝐾 + 1) MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7675rexbidva 3196 . 2 (𝜑 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(𝐾 + 1) MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
778, 76mpbird 248 1 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(𝐾 + 1) MonoAP 𝑓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  wss 3734  {csn 4336  cop 4342   class class class wbr 4811  cmpt 4890  ccnv 5278  dom cdm 5279  ran crn 5280  cima 5282  wf 6066  cfv 6070  (class class class)co 6846  𝑚 cmap 8064  cdom 8162  Fincfn 8164  cr 10192  1c1 10194   + caddc 10196   < clt 10332  cle 10333  cn 11278  2c2 11331  0cn0 11542  cuz 11891  ...cfz 12538  chash 13326  APcvdwa 15962   MonoAP cvdwm 15963   PolyAP cvdwp 15964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-er 7951  df-map 8066  df-pm 8067  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-card 9020  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-nn 11279  df-2 11339  df-n0 11543  df-xnn0 11615  df-z 11629  df-uz 11892  df-rp 12034  df-fz 12539  df-hash 13327  df-vdwap 15965  df-vdwmc 15966  df-vdwpc 15967
This theorem is referenced by:  vdwlem13  15990
  Copyright terms: Public domain W3C validator