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Theorem vdwlem11 17025
Description: Lemma for vdw 17028. (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 14392 . . . . 5 (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ0)
51, 4syl 17 . . . 4 (𝜑 → (♯‘𝑅) ∈ ℕ0)
6 nn0p1nn 12563 . . . 4 ((♯‘𝑅) ∈ ℕ0 → ((♯‘𝑅) + 1) ∈ ℕ)
75, 6syl 17 . . 3 (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
81, 2, 3, 7vdwlem10 17024 . 2 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))(⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
91adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑅 ∈ Fin)
10 ovex 7464 . . . . . . 7 (1...𝑛) ∈ V
11 elmapg 8878 . . . . . . 7 ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → (𝑓 ∈ (𝑅m (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
129, 10, 11sylancl 586 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑓 ∈ (𝑅m (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
1312biimpa 476 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅m (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅)
145nn0red 12586 . . . . . . . . . . 11 (𝜑 → (♯‘𝑅) ∈ ℝ)
1514ltp1d 12196 . . . . . . . . . 10 (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
16 peano2re 11432 . . . . . . . . . . . 12 ((♯‘𝑅) ∈ ℝ → ((♯‘𝑅) + 1) ∈ ℝ)
1714, 16syl 17 . . . . . . . . . . 11 (𝜑 → ((♯‘𝑅) + 1) ∈ ℝ)
1814, 17ltnled 11406 . . . . . . . . . 10 (𝜑 → ((♯‘𝑅) < ((♯‘𝑅) + 1) ↔ ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
1915, 18mpbid 232 . . . . . . . . 9 (𝜑 → ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅))
2019adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅))
21 eluz2nn 12922 . . . . . . . . . . . . 13 (𝐾 ∈ (ℤ‘2) → 𝐾 ∈ ℕ)
222, 21syl 17 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℕ)
2322adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ)
2423nnnn0d 12585 . . . . . . . . . 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)
267adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((♯‘𝑅) + 1) ∈ ℕ)
27 eqid 2735 . . . . . . . . . 10 (1...((♯‘𝑅) + 1)) = (1...((♯‘𝑅) + 1))
2810, 24, 25, 26, 27vdwpc 17014 . . . . . . . . 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 730 . . . . . . . . . . . . . 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 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)
3125ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . . 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 6102 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ⊆ dom 𝑓
3432, 33sstrdi 4008 . . . . . . . . . . . . . . . . . . . . . 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 6755 . . . . . . . . . . . . . . . . . . . . 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 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 12270 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ℕ ∈ V
40 ovex 7464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (1...((♯‘𝑅) + 1)) ∈ V
4139, 40elmap 8910 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 ∈ (ℕ ↑m (1...((♯‘𝑅) + 1))) ↔ 𝑑:(1...((♯‘𝑅) + 1))⟶ℕ)
4238, 41sylib 218 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...((♯‘𝑅) + 1))))) → 𝑑:(1...((♯‘𝑅) + 1))⟶ℕ)
4342ffvelcdmda 7104 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → (𝑑𝑖) ∈ ℕ)
4437, 43nnaddcld 12316 . . . . . . . . . . . . . . . . . . . . . . 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 17009 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐾 ∈ ℕ ∧ (𝑎 + (𝑑𝑖)) ∈ ℕ ∧ (𝑑𝑖) ∈ ℕ) → (𝑎 + (𝑑𝑖)) ∈ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
4636, 44, 43, 45syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . 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 ffvelcdm 7101 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑛)⟶𝑅 ∧ (𝑎 + (𝑑𝑖)) ∈ (1...𝑛)) → (𝑓‘(𝑎 + (𝑑𝑖))) ∈ 𝑅)
5130, 49, 50syl6an 684 . . . . . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))
5554fmpt 7130 . . . . . . . . . . . . . . . 16 (∀𝑖 ∈ (1...((♯‘𝑅) + 1))(𝑓‘(𝑎 + (𝑑𝑖))) ∈ 𝑅 ↔ (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))):(1...((♯‘𝑅) + 1))⟶𝑅)
5653, 55sylib 218 . . . . . . . . . . . . . . 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 6745 . . . . . . . . . . . . . 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 9039 . . . . . . . . . . . . . 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)
6029, 57ssfid 9299 . . . . . . . . . . . . . 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)
61 hashdom 14415 . . . . . . . . . . . . . 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))
6260, 29, 61syl2anc 584 . . . . . . . . . . . . 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))
6359, 62mpbird 257 . . . . . . . . . . . 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))
64 breq1 5151 . . . . . . . . . . . 12 ((♯‘ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = ((♯‘𝑅) + 1) → ((♯‘ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) ≤ (♯‘𝑅) ↔ ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
6563, 64syl5ibcom 245 . . . . . . . . . . 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)))
6665expimpd 453 . . . . . . . . . 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)))
6766rexlimdvva 3211 . . . . . . . . 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)))
6828, 67sylbid 240 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
6920, 68mtod 198 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓)
70 biorf 936 . . . . . . 7 (¬ ⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7169, 70syl 17 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7271anassrs 467 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓:(1...𝑛)⟶𝑅) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7313, 72syldan 591 . . . 4 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅m (1...𝑛))) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7473ralbidva 3174 . . 3 ((𝜑𝑛 ∈ ℕ) → (∀𝑓 ∈ (𝑅m (1...𝑛))(𝐾 + 1) MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅m (1...𝑛))(⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7574rexbidva 3175 . 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)))
768, 75mpbird 257 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 206  wa 395  wo 847   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963  {csn 4631  cop 4637   class class class wbr 5148  cmpt 5231  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  cdom 8982  Fincfn 8984  cr 11152  1c1 11154   + caddc 11156   < clt 11293  cle 11294  cn 12264  2c2 12319  0cn0 12524  cuz 12876  ...cfz 13544  chash 14366  APcvdwa 16999   MonoAP cvdwm 17000   PolyAP cvdwp 17001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-hash 14367  df-vdwap 17002  df-vdwmc 17003  df-vdwpc 17004
This theorem is referenced by:  vdwlem13  17027
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