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Theorem vdwlem11 16317
 Description: Lemma for vdw 16320. (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 13713 . . . . 5 (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ0)
51, 4syl 17 . . . 4 (𝜑 → (♯‘𝑅) ∈ ℕ0)
6 nn0p1nn 11924 . . . 4 ((♯‘𝑅) ∈ ℕ0 → ((♯‘𝑅) + 1) ∈ ℕ)
75, 6syl 17 . . 3 (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
81, 2, 3, 7vdwlem10 16316 . 2 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))(⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
91adantr 484 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑅 ∈ Fin)
10 ovex 7168 . . . . . . 7 (1...𝑛) ∈ V
11 elmapg 8402 . . . . . . 7 ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → (𝑓 ∈ (𝑅m (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
129, 10, 11sylancl 589 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑓 ∈ (𝑅m (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
1312biimpa 480 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅m (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅)
145nn0red 11944 . . . . . . . . . . 11 (𝜑 → (♯‘𝑅) ∈ ℝ)
1514ltp1d 11559 . . . . . . . . . 10 (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
16 peano2re 10802 . . . . . . . . . . . 12 ((♯‘𝑅) ∈ ℝ → ((♯‘𝑅) + 1) ∈ ℝ)
1714, 16syl 17 . . . . . . . . . . 11 (𝜑 → ((♯‘𝑅) + 1) ∈ ℝ)
1814, 17ltnled 10776 . . . . . . . . . 10 (𝜑 → ((♯‘𝑅) < ((♯‘𝑅) + 1) ↔ ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
1915, 18mpbid 235 . . . . . . . . 9 (𝜑 → ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅))
2019adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅))
21 eluz2nn 12272 . . . . . . . . . . . . 13 (𝐾 ∈ (ℤ‘2) → 𝐾 ∈ ℕ)
222, 21syl 17 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℕ)
2322adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ)
2423nnnn0d 11943 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ0)
25 simprr 772 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝑓:(1...𝑛)⟶𝑅)
267adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((♯‘𝑅) + 1) ∈ ℕ)
27 eqid 2798 . . . . . . . . . 10 (1...((♯‘𝑅) + 1)) = (1...((♯‘𝑅) + 1))
2810, 24, 25, 26, 27vdwpc 16306 . . . . . . . . 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 729 . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . . . . . . 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 5916 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ⊆ dom 𝑓
3432, 33sstrdi 3927 . . . . . . . . . . . . . . . . . . . . . 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 6503 . . . . . . . . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → 𝐾 ∈ ℕ)
37 simplrl 776 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → 𝑎 ∈ ℕ)
38 simprr 772 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...((♯‘𝑅) + 1))))) → 𝑑 ∈ (ℕ ↑m (1...((♯‘𝑅) + 1))))
39 nnex 11631 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ℕ ∈ V
40 ovex 7168 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (1...((♯‘𝑅) + 1)) ∈ V
4139, 40elmap 8418 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 ∈ (ℕ ↑m (1...((♯‘𝑅) + 1))) ↔ 𝑑:(1...((♯‘𝑅) + 1))⟶ℕ)
4238, 41sylib 221 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...((♯‘𝑅) + 1))))) → 𝑑:(1...((♯‘𝑅) + 1))⟶ℕ)
4342ffvelrnda 6828 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → (𝑑𝑖) ∈ ℕ)
4437, 43nnaddcld 11677 . . . . . . . . . . . . . . . . . . . . . . 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 16301 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐾 ∈ ℕ ∧ (𝑎 + (𝑑𝑖)) ∈ ℕ ∧ (𝑑𝑖) ∈ ℕ) → (𝑎 + (𝑑𝑖)) ∈ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
4636, 44, 43, 45syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . 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 3916 . . . . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . . . . . 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 6826 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑛)⟶𝑅 ∧ (𝑎 + (𝑑𝑖)) ∈ (1...𝑛)) → (𝑓‘(𝑎 + (𝑑𝑖))) ∈ 𝑅)
5130, 49, 50syl6an 683 . . . . . . . . . . . . . . . . . 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 3144 . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))
5554fmpt 6851 . . . . . . . . . . . . . . . 16 (∀𝑖 ∈ (1...((♯‘𝑅) + 1))(𝑓‘(𝑎 + (𝑑𝑖))) ∈ 𝑅 ↔ (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))):(1...((♯‘𝑅) + 1))⟶𝑅)
5653, 55sylib 221 . . . . . . . . . . . . . . 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 6494 . . . . . . . . . . . . . 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 8538 . . . . . . . . . . . . . 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 8725 . . . . . . . . . . . . . 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 13736 . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . 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 260 . . . . . . . . . . . 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 5033 . . . . . . . . . . . 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 248 . . . . . . . . . . 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 457 . . . . . . . . . 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 3253 . . . . . . . . 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 243 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
6920, 68mtod 201 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓)
70 biorf 934 . . . . . . 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 471 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓:(1...𝑛)⟶𝑅) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7313, 72syldan 594 . . . 4 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅m (1...𝑛))) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7473ralbidva 3161 . . 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 3255 . 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 260 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 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  {csn 4525  ⟨cop 4531   class class class wbr 5030   ↦ cmpt 5110  ◡ccnv 5518  dom cdm 5519  ran crn 5520   “ cima 5522  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↑m cmap 8389   ≼ cdom 8490  Fincfn 8492  ℝcr 10525  1c1 10527   + caddc 10529   < clt 10664   ≤ cle 10665  ℕcn 11625  2c2 11680  ℕ0cn0 11885  ℤ≥cuz 12231  ...cfz 12885  ♯chash 13686  APcvdwa 16291   MonoAP cvdwm 16292   PolyAP cvdwp 16293 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-hash 13687  df-vdwap 16294  df-vdwmc 16295  df-vdwpc 16296 This theorem is referenced by:  vdwlem13  16319
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