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Theorem vdwlem10 15637
Description: Lemma for vdw 15641. Set up secondary induction on 𝑀. (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)
vdwlem10.m (𝜑𝑀 ∈ ℕ)
Assertion
Ref Expression
vdwlem10 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
Distinct variable groups:   𝜑,𝑛,𝑓   𝑓,𝑠,𝐾,𝑛   𝑓,𝑀,𝑛   𝑅,𝑓,𝑛,𝑠   𝜑,𝑓
Allowed substitution hints:   𝜑(𝑠)   𝑀(𝑠)

Proof of Theorem vdwlem10
Dummy variables 𝑎 𝑐 𝑑 𝑔 𝑘 𝑚 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdwlem10.m . 2 (𝜑𝑀 ∈ ℕ)
2 opeq1 4377 . . . . . . 7 (𝑥 = 1 → ⟨𝑥, 𝐾⟩ = ⟨1, 𝐾⟩)
32breq1d 4633 . . . . . 6 (𝑥 = 1 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨1, 𝐾⟩ PolyAP 𝑓))
43orbi1d 738 . . . . 5 (𝑥 = 1 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
54rexralbidv 3053 . . . 4 (𝑥 = 1 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
65imbi2d 330 . . 3 (𝑥 = 1 → ((𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) ↔ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))))
7 opeq1 4377 . . . . . . 7 (𝑥 = 𝑚 → ⟨𝑥, 𝐾⟩ = ⟨𝑚, 𝐾⟩)
87breq1d 4633 . . . . . 6 (𝑥 = 𝑚 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑓))
98orbi1d 738 . . . . 5 (𝑥 = 𝑚 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
109rexralbidv 3053 . . . 4 (𝑥 = 𝑚 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
1110imbi2d 330 . . 3 (𝑥 = 𝑚 → ((𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) ↔ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))))
12 opeq1 4377 . . . . . . 7 (𝑥 = (𝑚 + 1) → ⟨𝑥, 𝐾⟩ = ⟨(𝑚 + 1), 𝐾⟩)
1312breq1d 4633 . . . . . 6 (𝑥 = (𝑚 + 1) → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓))
1413orbi1d 738 . . . . 5 (𝑥 = (𝑚 + 1) → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
1514rexralbidv 3053 . . . 4 (𝑥 = (𝑚 + 1) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
1615imbi2d 330 . . 3 (𝑥 = (𝑚 + 1) → ((𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) ↔ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))))
17 opeq1 4377 . . . . . . 7 (𝑥 = 𝑀 → ⟨𝑥, 𝐾⟩ = ⟨𝑀, 𝐾⟩)
1817breq1d 4633 . . . . . 6 (𝑥 = 𝑀 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑀, 𝐾⟩ PolyAP 𝑓))
1918orbi1d 738 . . . . 5 (𝑥 = 𝑀 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
2019rexralbidv 3053 . . . 4 (𝑥 = 𝑀 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
2120imbi2d 330 . . 3 (𝑥 = 𝑀 → ((𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) ↔ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))))
22 vdw.r . . . . . 6 (𝜑𝑅 ∈ Fin)
23 vdwlem9.s . . . . . 6 (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
24 oveq1 6622 . . . . . . . . 9 (𝑠 = 𝑅 → (𝑠𝑚 (1...𝑛)) = (𝑅𝑚 (1...𝑛)))
2524raleqdv 3137 . . . . . . . 8 (𝑠 = 𝑅 → (∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
2625rexbidv 3047 . . . . . . 7 (𝑠 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
2726rspcv 3295 . . . . . 6 (𝑅 ∈ Fin → (∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
2822, 23, 27sylc 65 . . . . 5 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
29 oveq2 6623 . . . . . . . 8 (𝑛 = 𝑤 → (1...𝑛) = (1...𝑤))
3029oveq2d 6631 . . . . . . 7 (𝑛 = 𝑤 → (𝑅𝑚 (1...𝑛)) = (𝑅𝑚 (1...𝑤)))
3130raleqdv 3137 . . . . . 6 (𝑛 = 𝑤 → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑓))
3231cbvrexv 3164 . . . . 5 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑓)
3328, 32sylib 208 . . . 4 (𝜑 → ∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑓)
34 breq2 4627 . . . . . . 7 (𝑓 = 𝑔 → (𝐾 MonoAP 𝑓𝐾 MonoAP 𝑔))
3534cbvralv 3163 . . . . . 6 (∀𝑓 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑔)
36 2nn 11145 . . . . . . . 8 2 ∈ ℕ
37 simpr 477 . . . . . . . 8 ((𝜑𝑤 ∈ ℕ) → 𝑤 ∈ ℕ)
38 nnmulcl 11003 . . . . . . . 8 ((2 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (2 · 𝑤) ∈ ℕ)
3936, 37, 38sylancr 694 . . . . . . 7 ((𝜑𝑤 ∈ ℕ) → (2 · 𝑤) ∈ ℕ)
4022adantr 481 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ℕ) → 𝑅 ∈ Fin)
41 ovex 6643 . . . . . . . . . . 11 (1...(2 · 𝑤)) ∈ V
42 elmapg 7830 . . . . . . . . . . 11 ((𝑅 ∈ Fin ∧ (1...(2 · 𝑤)) ∈ V) → (𝑓 ∈ (𝑅𝑚 (1...(2 · 𝑤))) ↔ 𝑓:(1...(2 · 𝑤))⟶𝑅))
4340, 41, 42sylancl 693 . . . . . . . . . 10 ((𝜑𝑤 ∈ ℕ) → (𝑓 ∈ (𝑅𝑚 (1...(2 · 𝑤))) ↔ 𝑓:(1...(2 · 𝑤))⟶𝑅))
4443biimpa 501 . . . . . . . . 9 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓 ∈ (𝑅𝑚 (1...(2 · 𝑤)))) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
45 simplr 791 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
46 elfznn 12328 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1...𝑤) → 𝑦 ∈ ℕ)
4746adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ∈ ℕ)
4847nnred 10995 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ∈ ℝ)
49 simpllr 798 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℕ)
5049nnred 10995 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℝ)
51 elfzle2 12303 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1...𝑤) → 𝑦𝑤)
5251adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦𝑤)
5348, 50, 50, 52leadd1dd 10601 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ≤ (𝑤 + 𝑤))
5449nncnd 10996 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℂ)
55542timesd 11235 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) = (𝑤 + 𝑤))
5653, 55breqtrrd 4651 . . . . . . . . . . . . . . 15 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ≤ (2 · 𝑤))
5747, 49nnaddcld 11027 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ ℕ)
58 nnuz 11683 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
5957, 58syl6eleq 2708 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ (ℤ‘1))
6039ad2antrr 761 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) ∈ ℕ)
6160nnzd 11441 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) ∈ ℤ)
62 elfz5 12292 . . . . . . . . . . . . . . . 16 (((𝑦 + 𝑤) ∈ (ℤ‘1) ∧ (2 · 𝑤) ∈ ℤ) → ((𝑦 + 𝑤) ∈ (1...(2 · 𝑤)) ↔ (𝑦 + 𝑤) ≤ (2 · 𝑤)))
6359, 61, 62syl2anc 692 . . . . . . . . . . . . . . 15 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → ((𝑦 + 𝑤) ∈ (1...(2 · 𝑤)) ↔ (𝑦 + 𝑤) ≤ (2 · 𝑤)))
6456, 63mpbird 247 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ (1...(2 · 𝑤)))
6545, 64ffvelrnd 6326 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑓‘(𝑦 + 𝑤)) ∈ 𝑅)
66 oveq1 6622 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 + 𝑤) = (𝑦 + 𝑤))
6766fveq2d 6162 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑓‘(𝑥 + 𝑤)) = (𝑓‘(𝑦 + 𝑤)))
6867cbvmptv 4720 . . . . . . . . . . . . 13 (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) = (𝑦 ∈ (1...𝑤) ↦ (𝑓‘(𝑦 + 𝑤)))
6965, 68fmptd 6351 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅)
70 ovex 6643 . . . . . . . . . . . . . 14 (1...𝑤) ∈ V
71 elmapg 7830 . . . . . . . . . . . . . 14 ((𝑅 ∈ Fin ∧ (1...𝑤) ∈ V) → ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅𝑚 (1...𝑤)) ↔ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅))
7240, 70, 71sylancl 693 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ℕ) → ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅𝑚 (1...𝑤)) ↔ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅))
7372biimpar 502 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℕ) ∧ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅𝑚 (1...𝑤)))
7469, 73syldan 487 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅𝑚 (1...𝑤)))
75 breq2 4627 . . . . . . . . . . . 12 (𝑔 = (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) → (𝐾 MonoAP 𝑔𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤)))))
7675rspcv 3295 . . . . . . . . . . 11 ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅𝑚 (1...𝑤)) → (∀𝑔 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑔𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤)))))
7774, 76syl 17 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (∀𝑔 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑔𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤)))))
78 2nn0 11269 . . . . . . . . . . . . 13 2 ∈ ℕ0
79 vdwlem9.k . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ (ℤ‘2))
8079ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → 𝐾 ∈ (ℤ‘2))
81 eluznn0 11717 . . . . . . . . . . . . 13 ((2 ∈ ℕ0𝐾 ∈ (ℤ‘2)) → 𝐾 ∈ ℕ0)
8278, 80, 81sylancr 694 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → 𝐾 ∈ ℕ0)
8370, 82, 69vdwmc 15625 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐})))
8440ad2antrr 761 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}))) → 𝑅 ∈ Fin)
8580adantr 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}))) → 𝐾 ∈ (ℤ‘2))
86 simpllr 798 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}))) → 𝑤 ∈ ℕ)
87 simplr 791 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}))) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
88 vex 3193 . . . . . . . . . . . . . . . 16 𝑐 ∈ V
89 simprll 801 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}))) → 𝑎 ∈ ℕ)
90 simprlr 802 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}))) → 𝑑 ∈ ℕ)
91 simprr 795 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}))) → (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}))
9284, 85, 86, 87, 88, 89, 90, 91, 68vdwlem8 15635 . . . . . . . . . . . . . . 15 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}))) → ⟨1, 𝐾⟩ PolyAP 𝑓)
9392orcd 407 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}))) → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
9493expr 642 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}) → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
9594rexlimdvva 3033 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}) → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
9695exlimdv 1858 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}) → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
9783, 96sylbid 230 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
9877, 97syld 47 . . . . . . . . 9 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (∀𝑔 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑔 → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
9944, 98syldan 487 . . . . . . . 8 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓 ∈ (𝑅𝑚 (1...(2 · 𝑤)))) → (∀𝑔 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑔 → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
10099ralrimdva 2965 . . . . . . 7 ((𝜑𝑤 ∈ ℕ) → (∀𝑔 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑔 → ∀𝑓 ∈ (𝑅𝑚 (1...(2 · 𝑤)))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
101 oveq2 6623 . . . . . . . . . 10 (𝑛 = (2 · 𝑤) → (1...𝑛) = (1...(2 · 𝑤)))
102101oveq2d 6631 . . . . . . . . 9 (𝑛 = (2 · 𝑤) → (𝑅𝑚 (1...𝑛)) = (𝑅𝑚 (1...(2 · 𝑤))))
103102raleqdv 3137 . . . . . . . 8 (𝑛 = (2 · 𝑤) → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∀𝑓 ∈ (𝑅𝑚 (1...(2 · 𝑤)))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
104103rspcev 3299 . . . . . . 7 (((2 · 𝑤) ∈ ℕ ∧ ∀𝑓 ∈ (𝑅𝑚 (1...(2 · 𝑤)))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
10539, 100, 104syl6an 567 . . . . . 6 ((𝜑𝑤 ∈ ℕ) → (∀𝑔 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑔 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
10635, 105syl5bi 232 . . . . 5 ((𝜑𝑤 ∈ ℕ) → (∀𝑓 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
107106rexlimdva 3026 . . . 4 (𝜑 → (∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
10833, 107mpd 15 . . 3 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
109 breq2 4627 . . . . . . . . . 10 (𝑓 = 𝑔 → (⟨𝑚, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑔))
110 breq2 4627 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝐾 + 1) MonoAP 𝑓 ↔ (𝐾 + 1) MonoAP 𝑔))
111109, 110orbi12d 745 . . . . . . . . 9 (𝑓 = 𝑔 → ((⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)))
112111cbvralv 3163 . . . . . . . 8 (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∀𝑔 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
11330raleqdv 3137 . . . . . . . 8 (𝑛 = 𝑤 → (∀𝑔 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) ↔ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)))
114112, 113syl5bb 272 . . . . . . 7 (𝑛 = 𝑤 → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)))
115114cbvrexv 3164 . . . . . 6 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∃𝑤 ∈ ℕ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
11622ad2antrr 761 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → 𝑅 ∈ Fin)
117 fzfi 12727 . . . . . . . . . 10 (1...𝑤) ∈ Fin
118 mapfi 8222 . . . . . . . . . 10 ((𝑅 ∈ Fin ∧ (1...𝑤) ∈ Fin) → (𝑅𝑚 (1...𝑤)) ∈ Fin)
119116, 117, 118sylancl 693 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → (𝑅𝑚 (1...𝑤)) ∈ Fin)
12023ad2antrr 761 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
121 oveq2 6623 . . . . . . . . . . . . . 14 (𝑛 = 𝑣 → (1...𝑛) = (1...𝑣))
122121oveq2d 6631 . . . . . . . . . . . . 13 (𝑛 = 𝑣 → (𝑠𝑚 (1...𝑛)) = (𝑠𝑚 (1...𝑣)))
123122raleqdv 3137 . . . . . . . . . . . 12 (𝑛 = 𝑣 → (∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
124123cbvrexv 3164 . . . . . . . . . . 11 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑣 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑣))𝐾 MonoAP 𝑓)
125 oveq1 6622 . . . . . . . . . . . . 13 (𝑠 = (𝑅𝑚 (1...𝑤)) → (𝑠𝑚 (1...𝑣)) = ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣)))
126125raleqdv 3137 . . . . . . . . . . . 12 (𝑠 = (𝑅𝑚 (1...𝑤)) → (∀𝑓 ∈ (𝑠𝑚 (1...𝑣))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
127126rexbidv 3047 . . . . . . . . . . 11 (𝑠 = (𝑅𝑚 (1...𝑤)) → (∃𝑣 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑣))𝐾 MonoAP 𝑓 ↔ ∃𝑣 ∈ ℕ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
128124, 127syl5bb 272 . . . . . . . . . 10 (𝑠 = (𝑅𝑚 (1...𝑤)) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑣 ∈ ℕ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
129128rspcv 3295 . . . . . . . . 9 ((𝑅𝑚 (1...𝑤)) ∈ Fin → (∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓 → ∃𝑣 ∈ ℕ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
130119, 120, 129sylc 65 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → ∃𝑣 ∈ ℕ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)
131 simprll 801 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓))) → 𝑤 ∈ ℕ)
132 simprrl 803 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓))) → 𝑣 ∈ ℕ)
133 nnmulcl 11003 . . . . . . . . . . . . 13 ((2 ∈ ℕ ∧ 𝑣 ∈ ℕ) → (2 · 𝑣) ∈ ℕ)
13436, 133mpan 705 . . . . . . . . . . . 12 (𝑣 ∈ ℕ → (2 · 𝑣) ∈ ℕ)
135 nnmulcl 11003 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ ∧ (2 · 𝑣) ∈ ℕ) → (𝑤 · (2 · 𝑣)) ∈ ℕ)
136134, 135sylan2 491 . . . . . . . . . . 11 ((𝑤 ∈ ℕ ∧ 𝑣 ∈ ℕ) → (𝑤 · (2 · 𝑣)) ∈ ℕ)
137131, 132, 136syl2anc 692 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓))) → (𝑤 · (2 · 𝑣)) ∈ ℕ)
138 simp1l 1083 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → 𝜑)
139138, 22syl 17 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → 𝑅 ∈ Fin)
140138, 79syl 17 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → 𝐾 ∈ (ℤ‘2))
141138, 23syl 17 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
142 simp1r 1084 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → 𝑚 ∈ ℕ)
143 simp2ll 1126 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → 𝑤 ∈ ℕ)
144 simp2lr 1127 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
145 breq2 4627 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑘 → (⟨𝑚, 𝐾⟩ PolyAP 𝑔 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑘))
146 breq2 4627 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑘 → ((𝐾 + 1) MonoAP 𝑔 ↔ (𝐾 + 1) MonoAP 𝑘))
147145, 146orbi12d 745 . . . . . . . . . . . . . . 15 (𝑔 = 𝑘 → ((⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑘 ∨ (𝐾 + 1) MonoAP 𝑘)))
148147cbvralv 3163 . . . . . . . . . . . . . 14 (∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) ↔ ∀𝑘 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑘 ∨ (𝐾 + 1) MonoAP 𝑘))
149144, 148sylib 208 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → ∀𝑘 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑘 ∨ (𝐾 + 1) MonoAP 𝑘))
150 simp2rl 1128 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → 𝑣 ∈ ℕ)
151 simp2rr 1129 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)
152 simp3 1061 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣)))))
153 ovex 6643 . . . . . . . . . . . . . . 15 (1...(𝑤 · (2 · 𝑣))) ∈ V
154 elmapg 7830 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Fin ∧ (1...(𝑤 · (2 · 𝑣))) ∈ V) → ( ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣)))) ↔ :(1...(𝑤 · (2 · 𝑣)))⟶𝑅))
155139, 153, 154sylancl 693 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → ( ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣)))) ↔ :(1...(𝑤 · (2 · 𝑣)))⟶𝑅))
156152, 155mpbid 222 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → :(1...(𝑤 · (2 · 𝑣)))⟶𝑅)
157 oveq1 6622 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑢 → (𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))) = (𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))))
158157fveq2d 6162 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣)))) = (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))))
159158cbvmptv 4720 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1...𝑤) ↦ (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))))
160 oveq1 6622 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑧 → (𝑥 − 1) = (𝑧 − 1))
161160oveq1d 6630 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → ((𝑥 − 1) + 𝑣) = ((𝑧 − 1) + 𝑣))
162161oveq2d 6631 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (𝑤 · ((𝑥 − 1) + 𝑣)) = (𝑤 · ((𝑧 − 1) + 𝑣)))
163162oveq2d 6631 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))) = (𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))
164163fveq2d 6162 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))) = (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣)))))
165164mpteq2dv 4715 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
166159, 165syl5eq 2667 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑦 ∈ (1...𝑤) ↦ (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
167166cbvmptv 4720 . . . . . . . . . . . . 13 (𝑥 ∈ (1...𝑣) ↦ (𝑦 ∈ (1...𝑤) ↦ (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣)))))) = (𝑧 ∈ (1...𝑣) ↦ (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
168139, 140, 141, 142, 143, 149, 150, 151, 156, 167vdwlem9 15636 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))) → (⟨(𝑚 + 1), 𝐾⟩ PolyAP ∨ (𝐾 + 1) MonoAP ))
1691683expia 1264 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓))) → ( ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣)))) → (⟨(𝑚 + 1), 𝐾⟩ PolyAP ∨ (𝐾 + 1) MonoAP )))
170169ralrimiv 2961 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓))) → ∀ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))(⟨(𝑚 + 1), 𝐾⟩ PolyAP ∨ (𝐾 + 1) MonoAP ))
171 oveq2 6623 . . . . . . . . . . . . . 14 (𝑛 = (𝑤 · (2 · 𝑣)) → (1...𝑛) = (1...(𝑤 · (2 · 𝑣))))
172171oveq2d 6631 . . . . . . . . . . . . 13 (𝑛 = (𝑤 · (2 · 𝑣)) → (𝑅𝑚 (1...𝑛)) = (𝑅𝑚 (1...(𝑤 · (2 · 𝑣)))))
173172raleqdv 3137 . . . . . . . . . . . 12 (𝑛 = (𝑤 · (2 · 𝑣)) → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∀𝑓 ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
174 breq2 4627 . . . . . . . . . . . . . 14 (𝑓 = → (⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ↔ ⟨(𝑚 + 1), 𝐾⟩ PolyAP ))
175 breq2 4627 . . . . . . . . . . . . . 14 (𝑓 = → ((𝐾 + 1) MonoAP 𝑓 ↔ (𝐾 + 1) MonoAP ))
176174, 175orbi12d 745 . . . . . . . . . . . . 13 (𝑓 = → ((⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨(𝑚 + 1), 𝐾⟩ PolyAP ∨ (𝐾 + 1) MonoAP )))
177176cbvralv 3163 . . . . . . . . . . . 12 (∀𝑓 ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∀ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))(⟨(𝑚 + 1), 𝐾⟩ PolyAP ∨ (𝐾 + 1) MonoAP ))
178173, 177syl6bb 276 . . . . . . . . . . 11 (𝑛 = (𝑤 · (2 · 𝑣)) → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∀ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))(⟨(𝑚 + 1), 𝐾⟩ PolyAP ∨ (𝐾 + 1) MonoAP )))
179178rspcev 3299 . . . . . . . . . 10 (((𝑤 · (2 · 𝑣)) ∈ ℕ ∧ ∀ ∈ (𝑅𝑚 (1...(𝑤 · (2 · 𝑣))))(⟨(𝑚 + 1), 𝐾⟩ PolyAP ∨ (𝐾 + 1) MonoAP )) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
180137, 170, 179syl2anc 692 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓))) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
181180anassrs 679 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
182130, 181rexlimddv 3030 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
183182rexlimdvaa 3027 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (∃𝑤 ∈ ℕ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
184115, 183syl5bi 232 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
185184expcom 451 . . . 4 (𝑚 ∈ ℕ → (𝜑 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))))
186185a2d 29 . . 3 (𝑚 ∈ ℕ → ((𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) → (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))))
1876, 11, 16, 21, 108, 186nnind 10998 . 2 (𝑀 ∈ ℕ → (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
1881, 187mpcom 38 1 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wral 2908  wrex 2909  Vcvv 3190  wss 3560  {csn 4155  cop 4161   class class class wbr 4623  cmpt 4683  ccnv 5083  cima 5087  wf 5853  cfv 5857  (class class class)co 6615  𝑚 cmap 7817  Fincfn 7915  1c1 9897   + caddc 9899   · cmul 9901  cle 10035  cmin 10226  cn 10980  2c2 11030  0cn0 11252  cz 11337  cuz 11647  ...cfz 12284  APcvdwa 15612   MonoAP cvdwm 15613   PolyAP cvdwp 15614
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-rp 11793  df-fz 12285  df-hash 13074  df-vdwap 15615  df-vdwmc 15616  df-vdwpc 15617
This theorem is referenced by:  vdwlem11  15638
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