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Theorem vdwlem10 15532
 Description: Lemma for vdw 15536. 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 4340 . . . . . . 7 (𝑥 = 1 → ⟨𝑥, 𝐾⟩ = ⟨1, 𝐾⟩)
32breq1d 4593 . . . . . 6 (𝑥 = 1 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨1, 𝐾⟩ PolyAP 𝑓))
43orbi1d 735 . . . . 5 (𝑥 = 1 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
54rexralbidv 3040 . . . 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 329 . . 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 4340 . . . . . . 7 (𝑥 = 𝑚 → ⟨𝑥, 𝐾⟩ = ⟨𝑚, 𝐾⟩)
87breq1d 4593 . . . . . 6 (𝑥 = 𝑚 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑓))
98orbi1d 735 . . . . 5 (𝑥 = 𝑚 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
109rexralbidv 3040 . . . 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 329 . . 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 4340 . . . . . . 7 (𝑥 = (𝑚 + 1) → ⟨𝑥, 𝐾⟩ = ⟨(𝑚 + 1), 𝐾⟩)
1312breq1d 4593 . . . . . 6 (𝑥 = (𝑚 + 1) → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓))
1413orbi1d 735 . . . . 5 (𝑥 = (𝑚 + 1) → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
1514rexralbidv 3040 . . . 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 329 . . 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 4340 . . . . . . 7 (𝑥 = 𝑀 → ⟨𝑥, 𝐾⟩ = ⟨𝑀, 𝐾⟩)
1817breq1d 4593 . . . . . 6 (𝑥 = 𝑀 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑀, 𝐾⟩ PolyAP 𝑓))
1918orbi1d 735 . . . . 5 (𝑥 = 𝑀 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
2019rexralbidv 3040 . . . 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 329 . . 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 6556 . . . . . . . . 9 (𝑠 = 𝑅 → (𝑠𝑚 (1...𝑛)) = (𝑅𝑚 (1...𝑛)))
2524raleqdv 3121 . . . . . . . 8 (𝑠 = 𝑅 → (∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
2625rexbidv 3034 . . . . . . 7 (𝑠 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
2726rspcv 3278 . . . . . 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 63 . . . . 5 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
29 oveq2 6557 . . . . . . . 8 (𝑛 = 𝑤 → (1...𝑛) = (1...𝑤))
3029oveq2d 6565 . . . . . . 7 (𝑛 = 𝑤 → (𝑅𝑚 (1...𝑛)) = (𝑅𝑚 (1...𝑤)))
3130raleqdv 3121 . . . . . 6 (𝑛 = 𝑤 → (∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑓))
3231cbvrexv 3148 . . . . 5 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑓)
3328, 32sylib 207 . . . 4 (𝜑 → ∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑓)
34 breq2 4587 . . . . . . 7 (𝑓 = 𝑔 → (𝐾 MonoAP 𝑓𝐾 MonoAP 𝑔))
3534cbvralv 3147 . . . . . 6 (∀𝑓 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑔)
36 2nn 11062 . . . . . . . 8 2 ∈ ℕ
37 simpr 476 . . . . . . . 8 ((𝜑𝑤 ∈ ℕ) → 𝑤 ∈ ℕ)
38 nnmulcl 10920 . . . . . . . 8 ((2 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (2 · 𝑤) ∈ ℕ)
3936, 37, 38sylancr 694 . . . . . . 7 ((𝜑𝑤 ∈ ℕ) → (2 · 𝑤) ∈ ℕ)
4022adantr 480 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ℕ) → 𝑅 ∈ Fin)
41 ovex 6577 . . . . . . . . . . 11 (1...(2 · 𝑤)) ∈ V
42 elmapg 7757 . . . . . . . . . . 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 500 . . . . . . . . 9 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓 ∈ (𝑅𝑚 (1...(2 · 𝑤)))) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
45 simplr 788 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
46 elfznn 12241 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1...𝑤) → 𝑦 ∈ ℕ)
4746adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ∈ ℕ)
4847nnred 10912 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ∈ ℝ)
49 simpllr 795 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℕ)
5049nnred 10912 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℝ)
51 elfzle2 12216 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1...𝑤) → 𝑦𝑤)
5251adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦𝑤)
5348, 50, 50, 52leadd1dd 10520 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ≤ (𝑤 + 𝑤))
5449nncnd 10913 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℂ)
55542timesd 11152 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) = (𝑤 + 𝑤))
5653, 55breqtrrd 4611 . . . . . . . . . . . . . . 15 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ≤ (2 · 𝑤))
5747, 49nnaddcld 10944 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ ℕ)
58 nnuz 11599 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
5957, 58syl6eleq 2698 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ (ℤ‘1))
6039ad2antrr 758 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) ∈ ℕ)
6160nnzd 11357 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) ∈ ℤ)
62 elfz5 12205 . . . . . . . . . . . . . . . 16 (((𝑦 + 𝑤) ∈ (ℤ‘1) ∧ (2 · 𝑤) ∈ ℤ) → ((𝑦 + 𝑤) ∈ (1...(2 · 𝑤)) ↔ (𝑦 + 𝑤) ≤ (2 · 𝑤)))
6359, 61, 62syl2anc 691 . . . . . . . . . . . . . . 15 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → ((𝑦 + 𝑤) ∈ (1...(2 · 𝑤)) ↔ (𝑦 + 𝑤) ≤ (2 · 𝑤)))
6456, 63mpbird 246 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ (1...(2 · 𝑤)))
6545, 64ffvelrnd 6268 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑓‘(𝑦 + 𝑤)) ∈ 𝑅)
66 oveq1 6556 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝑥 + 𝑤) = (𝑦 + 𝑤))
6766fveq2d 6107 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑓‘(𝑥 + 𝑤)) = (𝑓‘(𝑦 + 𝑤)))
6867cbvmptv 4678 . . . . . . . . . . . . 13 (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) = (𝑦 ∈ (1...𝑤) ↦ (𝑓‘(𝑦 + 𝑤)))
6965, 68fmptd 6292 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅)
70 ovex 6577 . . . . . . . . . . . . . 14 (1...𝑤) ∈ V
71 elmapg 7757 . . . . . . . . . . . . . 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 501 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℕ) ∧ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅𝑚 (1...𝑤)))
7469, 73syldan 486 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅𝑚 (1...𝑤)))
75 breq2 4587 . . . . . . . . . . . 12 (𝑔 = (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) → (𝐾 MonoAP 𝑔𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤)))))
7675rspcv 3278 . . . . . . . . . . 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 11186 . . . . . . . . . . . . 13 2 ∈ ℕ0
79 vdwlem9.k . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ (ℤ‘2))
8079ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → 𝐾 ∈ (ℤ‘2))
81 eluznn0 11633 . . . . . . . . . . . . 13 ((2 ∈ ℕ0𝐾 ∈ (ℤ‘2)) → 𝐾 ∈ ℕ0)
8278, 80, 81sylancr 694 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → 𝐾 ∈ ℕ0)
8370, 82, 69vdwmc 15520 . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 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 795 . . . . . . . . . . . . . . . 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 788 . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . . 16 𝑐 ∈ V
89 simprll 798 . . . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . . . 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 792 . . . . . . . . . . . . . . . 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 15530 . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . 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 641 . . . . . . . . . . . . 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 3020 . . . . . . . . . . . 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 1848 . . . . . . . . . . 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 229 . . . . . . . . . 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 46 . . . . . . . . 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 486 . . . . . . . 8 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓 ∈ (𝑅𝑚 (1...(2 · 𝑤)))) → (∀𝑔 ∈ (𝑅𝑚 (1...𝑤))𝐾 MonoAP 𝑔 → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
10099ralrimdva 2952 . . . . . . 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 6557 . . . . . . . . . 10 (𝑛 = (2 · 𝑤) → (1...𝑛) = (1...(2 · 𝑤)))
102101oveq2d 6565 . . . . . . . . 9 (𝑛 = (2 · 𝑤) → (𝑅𝑚 (1...𝑛)) = (𝑅𝑚 (1...(2 · 𝑤))))
103102raleqdv 3121 . . . . . . . 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 3282 . . . . . . 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 566 . . . . . 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 231 . . . . 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 3013 . . . 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 4587 . . . . . . . . . 10 (𝑓 = 𝑔 → (⟨𝑚, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑔))
110 breq2 4587 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝐾 + 1) MonoAP 𝑓 ↔ (𝐾 + 1) MonoAP 𝑔))
111109, 110orbi12d 742 . . . . . . . . 9 (𝑓 = 𝑔 → ((⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)))
112111cbvralv 3147 . . . . . . . 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 3121 . . . . . . . 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 271 . . . . . . 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 3148 . . . . . 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 758 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → 𝑅 ∈ Fin)
117 fzfi 12633 . . . . . . . . . 10 (1...𝑤) ∈ Fin
118 mapfi 8145 . . . . . . . . . 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 758 . . . . . . . . 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 6557 . . . . . . . . . . . . . 14 (𝑛 = 𝑣 → (1...𝑛) = (1...𝑣))
122121oveq2d 6565 . . . . . . . . . . . . 13 (𝑛 = 𝑣 → (𝑠𝑚 (1...𝑛)) = (𝑠𝑚 (1...𝑣)))
123122raleqdv 3121 . . . . . . . . . . . 12 (𝑛 = 𝑣 → (∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
124123cbvrexv 3148 . . . . . . . . . . 11 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑣 ∈ ℕ ∀𝑓 ∈ (𝑠𝑚 (1...𝑣))𝐾 MonoAP 𝑓)
125 oveq1 6556 . . . . . . . . . . . . 13 (𝑠 = (𝑅𝑚 (1...𝑤)) → (𝑠𝑚 (1...𝑣)) = ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣)))
126125raleqdv 3121 . . . . . . . . . . . 12 (𝑠 = (𝑅𝑚 (1...𝑤)) → (∀𝑓 ∈ (𝑠𝑚 (1...𝑣))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ ((𝑅𝑚 (1...𝑤)) ↑𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
127126rexbidv 3034 . . . . . . . . . . 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 271 . . . . . . . . . 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 3278 . . . . . . . . 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 63 . . . . . . . 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 798 . . . . . . . . . . 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 800 . . . . . . . . . . 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 10920 . . . . . . . . . . . . 13 ((2 ∈ ℕ ∧ 𝑣 ∈ ℕ) → (2 · 𝑣) ∈ ℕ)
13436, 133mpan 702 . . . . . . . . . . . 12 (𝑣 ∈ ℕ → (2 · 𝑣) ∈ ℕ)
135 nnmulcl 10920 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ ∧ (2 · 𝑣) ∈ ℕ) → (𝑤 · (2 · 𝑣)) ∈ ℕ)
136134, 135sylan2 490 . . . . . . . . . . 11 ((𝑤 ∈ ℕ ∧ 𝑣 ∈ ℕ) → (𝑤 · (2 · 𝑣)) ∈ ℕ)
137131, 132, 136syl2anc 691 . . . . . . . . . 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 1078 . . . . . . . . . . . . . 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 1079 . . . . . . . . . . . . 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 1121 . . . . . . . . . . . . 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 1122 . . . . . . . . . . . . . 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 4587 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑘 → (⟨𝑚, 𝐾⟩ PolyAP 𝑔 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑘))
146 breq2 4587 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑘 → ((𝐾 + 1) MonoAP 𝑔 ↔ (𝐾 + 1) MonoAP 𝑘))
147145, 146orbi12d 742 . . . . . . . . . . . . . . 15 (𝑔 = 𝑘 → ((⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑘 ∨ (𝐾 + 1) MonoAP 𝑘)))
148147cbvralv 3147 . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . 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 1123 . . . . . . . . . . . . 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 1124 . . . . . . . . . . . . 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 1056 . . . . . . . . . . . . . 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 6577 . . . . . . . . . . . . . . 15 (1...(𝑤 · (2 · 𝑣))) ∈ V
154 elmapg 7757 . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . 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 6556 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑢 → (𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))) = (𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))))
158157fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣)))) = (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))))
159158cbvmptv 4678 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1...𝑤) ↦ (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))))
160 oveq1 6556 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑧 → (𝑥 − 1) = (𝑧 − 1))
161160oveq1d 6564 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → ((𝑥 − 1) + 𝑣) = ((𝑧 − 1) + 𝑣))
162161oveq2d 6565 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (𝑤 · ((𝑥 − 1) + 𝑣)) = (𝑤 · ((𝑧 − 1) + 𝑣)))
163162oveq2d 6565 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))) = (𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))
164163fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))) = (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣)))))
165164mpteq2dv 4673 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
166159, 165syl5eq 2656 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑦 ∈ (1...𝑤) ↦ (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
167166cbvmptv 4678 . . . . . . . . . . . . 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 15531 . . . . . . . . . . . 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 1259 . . . . . . . . . . 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 2948 . . . . . . . . . 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 6557 . . . . . . . . . . . . . 14 (𝑛 = (𝑤 · (2 · 𝑣)) → (1...𝑛) = (1...(𝑤 · (2 · 𝑣))))
172171oveq2d 6565 . . . . . . . . . . . . 13 (𝑛 = (𝑤 · (2 · 𝑣)) → (𝑅𝑚 (1...𝑛)) = (𝑅𝑚 (1...(𝑤 · (2 · 𝑣)))))
173172raleqdv 3121 . . . . . . . . . . . 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 4587 . . . . . . . . . . . . . 14 (𝑓 = → (⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ↔ ⟨(𝑚 + 1), 𝐾⟩ PolyAP ))
175 breq2 4587 . . . . . . . . . . . . . 14 (𝑓 = → ((𝐾 + 1) MonoAP 𝑓 ↔ (𝐾 + 1) MonoAP ))
176174, 175orbi12d 742 . . . . . . . . . . . . 13 (𝑓 = → ((⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨(𝑚 + 1), 𝐾⟩ PolyAP ∨ (𝐾 + 1) MonoAP )))
177176cbvralv 3147 . . . . . . . . . . . 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 275 . . . . . . . . . . 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 3282 . . . . . . . . . 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 691 . . . . . . . . 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 678 . . . . . . . 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 3017 . . . . . . 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 3014 . . . . . 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 231 . . . . 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 450 . . . 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 10915 . 2 (𝑀 ∈ ℕ → (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅𝑚 (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
1881, 187mpcom 37 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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ◡ccnv 5037   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Fincfn 7841  1c1 9816   + caddc 9818   · cmul 9820   ≤ cle 9954   − cmin 10145  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  APcvdwa 15507   MonoAP cvdwm 15508   PolyAP cvdwp 15509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-hash 12980  df-vdwap 15510  df-vdwmc 15511  df-vdwpc 15512 This theorem is referenced by:  vdwlem11  15533
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