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Theorem vdwlem11 15614
Description: Lemma for vdw 15617. (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 13084 . . . . 5 (𝑅 ∈ Fin → (#‘𝑅) ∈ ℕ0)
51, 4syl 17 . . . 4 (𝜑 → (#‘𝑅) ∈ ℕ0)
6 nn0p1nn 11277 . . . 4 ((#‘𝑅) ∈ ℕ0 → ((#‘𝑅) + 1) ∈ ℕ)
75, 6syl 17 . . 3 (𝜑 → ((#‘𝑅) + 1) ∈ ℕ)
81, 2, 3, 7vdwlem10 15613 . 2 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
91adantr 481 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑅 ∈ Fin)
10 ovex 6633 . . . . . . 7 (1...𝑛) ∈ V
11 elmapg 7816 . . . . . . 7 ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → (𝑓 ∈ (𝑅𝑚 (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
129, 10, 11sylancl 693 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑓 ∈ (𝑅𝑚 (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
1312biimpa 501 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅𝑚 (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅)
145nn0red 11297 . . . . . . . . . . 11 (𝜑 → (#‘𝑅) ∈ ℝ)
1514ltp1d 10899 . . . . . . . . . 10 (𝜑 → (#‘𝑅) < ((#‘𝑅) + 1))
16 peano2re 10154 . . . . . . . . . . . 12 ((#‘𝑅) ∈ ℝ → ((#‘𝑅) + 1) ∈ ℝ)
1714, 16syl 17 . . . . . . . . . . 11 (𝜑 → ((#‘𝑅) + 1) ∈ ℝ)
1814, 17ltnled 10129 . . . . . . . . . 10 (𝜑 → ((#‘𝑅) < ((#‘𝑅) + 1) ↔ ¬ ((#‘𝑅) + 1) ≤ (#‘𝑅)))
1915, 18mpbid 222 . . . . . . . . 9 (𝜑 → ¬ ((#‘𝑅) + 1) ≤ (#‘𝑅))
2019adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ((#‘𝑅) + 1) ≤ (#‘𝑅))
21 eluz2nn 11670 . . . . . . . . . . . . 13 (𝐾 ∈ (ℤ‘2) → 𝐾 ∈ ℕ)
222, 21syl 17 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℕ)
2322adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ)
2423nnnn0d 11296 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ0)
25 simprr 795 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝑓:(1...𝑛)⟶𝑅)
267adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((#‘𝑅) + 1) ∈ ℕ)
27 eqid 2626 . . . . . . . . . 10 (1...((#‘𝑅) + 1)) = (1...((#‘𝑅) + 1))
2810, 24, 25, 26, 27vdwpc 15603 . . . . . . . . 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 765 . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . . 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 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)))}))
32 cnvimass 5448 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 “ {(𝑓‘(𝑎 + (𝑑𝑖)))}) ⊆ dom 𝑓
3331, 32syl6ss 3600 . . . . . . . . . . . . . . . . . . . . . 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)
3425ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . 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)))})) → 𝑓:(1...𝑛)⟶𝑅)
35 fdm 6010 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓:(1...𝑛)⟶𝑅 → dom 𝑓 = (1...𝑛))
3634, 35syl 17 . . . . . . . . . . . . . . . . . . . . . 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)))})) → dom 𝑓 = (1...𝑛))
3733, 36sseqtrd 3625 . . . . . . . . . . . . . . . . . . . . 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))
3822ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → 𝐾 ∈ ℕ)
39 simplrl 799 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → 𝑎 ∈ ℕ)
40 simprr 795 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((#‘𝑅) + 1))))) → 𝑑 ∈ (ℕ ↑𝑚 (1...((#‘𝑅) + 1))))
41 nnex 10971 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ℕ ∈ V
42 ovex 6633 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (1...((#‘𝑅) + 1)) ∈ V
4341, 42elmap 7831 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 ∈ (ℕ ↑𝑚 (1...((#‘𝑅) + 1))) ↔ 𝑑:(1...((#‘𝑅) + 1))⟶ℕ)
4440, 43sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((#‘𝑅) + 1))))) → 𝑑:(1...((#‘𝑅) + 1))⟶ℕ)
4544ffvelrnda 6316 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → (𝑑𝑖) ∈ ℕ)
4639, 45nnaddcld 11012 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → (𝑎 + (𝑑𝑖)) ∈ ℕ)
47 vdwapid1 15598 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐾 ∈ ℕ ∧ (𝑎 + (𝑑𝑖)) ∈ ℕ ∧ (𝑑𝑖) ∈ ℕ) → (𝑎 + (𝑑𝑖)) ∈ ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)))
4838, 46, 45, 47syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . 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)))
4948adantr 481 . . . . . . . . . . . . . . . . . . . . 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)))
5037, 49sseldd 3589 . . . . . . . . . . . . . . . . . . . 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))
5150ex 450 . . . . . . . . . . . . . . . . . . 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)))
52 ffvelrn 6314 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑛)⟶𝑅 ∧ (𝑎 + (𝑑𝑖)) ∈ (1...𝑛)) → (𝑓‘(𝑎 + (𝑑𝑖))) ∈ 𝑅)
5330, 51, 52syl6an 567 . . . . . . . . . . . . . . . . . 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))
5453ralimdva 2961 . . . . . . . . . . . . . . . . 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))
5554imp 445 . . . . . . . . . . . . . . . 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)
56 eqid 2626 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) = (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))
5756fmpt 6338 . . . . . . . . . . . . . . . 16 (∀𝑖 ∈ (1...((#‘𝑅) + 1))(𝑓‘(𝑎 + (𝑑𝑖))) ∈ 𝑅 ↔ (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))):(1...((#‘𝑅) + 1))⟶𝑅)
5855, 57sylib 208 . . . . . . . . . . . . . . 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)
59 frn 6012 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))):(1...((#‘𝑅) + 1))⟶𝑅 → ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ⊆ 𝑅)
6058, 59syl 17 . . . . . . . . . . . . . 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)
61 ssdomg 7946 . . . . . . . . . . . . . 14 (𝑅 ∈ Fin → (ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ⊆ 𝑅 → ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ≼ 𝑅))
6229, 60, 61sylc 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)
63 ssfi 8125 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Fin ∧ ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ⊆ 𝑅) → ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖)))) ∈ Fin)
6429, 60, 63syl2anc 692 . . . . . . . . . . . . . 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)
65 hashdom 13105 . . . . . . . . . . . . . 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))
6664, 29, 65syl2anc 692 . . . . . . . . . . . . 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))
6762, 66mpbird 247 . . . . . . . . . . . 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))
68 breq1 4621 . . . . . . . . . . . 12 ((#‘ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) = ((#‘𝑅) + 1) → ((#‘ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑𝑖))))) ≤ (#‘𝑅) ↔ ((#‘𝑅) + 1) ≤ (#‘𝑅)))
6967, 68syl5ibcom 235 . . . . . . . . . . 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)))
7069expimpd 628 . . . . . . . . . 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)))
7170rexlimdvva 3036 . . . . . . . . 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)))
7228, 71sylbid 230 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((#‘𝑅) + 1) ≤ (#‘𝑅)))
7320, 72mtod 189 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓)
74 biorf 420 . . . . . . 7 (¬ ⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7573, 74syl 17 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7675anassrs 679 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓:(1...𝑛)⟶𝑅) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7713, 76syldan 487 . . . 4 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅𝑚 (1...𝑛))) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7877ralbidva 2984 . . 3 ((𝜑𝑛 ∈ ℕ) → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(𝐾 + 1) MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
7978rexbidva 3047 . 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)))
808, 79mpbird 247 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 196  wo 383  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  Vcvv 3191  wss 3560  {csn 4153  cop 4159   class class class wbr 4618  cmpt 4678  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082  wf 5846  cfv 5850  (class class class)co 6605  𝑚 cmap 7803  cdom 7898  Fincfn 7900  cr 9880  1c1 9882   + caddc 9884   < clt 10019  cle 10020  cn 10965  2c2 11015  0cn0 11237  cuz 11631  ...cfz 12265  #chash 13054  APcvdwa 15588   MonoAP cvdwm 15589   PolyAP cvdwp 15590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-xnn0 11309  df-z 11323  df-uz 11632  df-rp 11777  df-fz 12266  df-hash 13055  df-vdwap 15591  df-vdwmc 15592  df-vdwpc 15593
This theorem is referenced by:  vdwlem13  15616
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