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Theorem vdwlem10 16862
Description: Lemma for vdw 16866. 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 4830 . . . . . . 7 (𝑥 = 1 → ⟨𝑥, 𝐾⟩ = ⟨1, 𝐾⟩)
32breq1d 5115 . . . . . 6 (𝑥 = 1 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨1, 𝐾⟩ PolyAP 𝑓))
43orbi1d 915 . . . . 5 (𝑥 = 1 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
54rexralbidv 3214 . . . 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 340 . . 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 4830 . . . . . . 7 (𝑥 = 𝑚 → ⟨𝑥, 𝐾⟩ = ⟨𝑚, 𝐾⟩)
87breq1d 5115 . . . . . 6 (𝑥 = 𝑚 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑓))
98orbi1d 915 . . . . 5 (𝑥 = 𝑚 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
109rexralbidv 3214 . . . 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 340 . . 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 4830 . . . . . . 7 (𝑥 = (𝑚 + 1) → ⟨𝑥, 𝐾⟩ = ⟨(𝑚 + 1), 𝐾⟩)
1312breq1d 5115 . . . . . 6 (𝑥 = (𝑚 + 1) → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓))
1413orbi1d 915 . . . . 5 (𝑥 = (𝑚 + 1) → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
1514rexralbidv 3214 . . . 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 340 . . 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 4830 . . . . . . 7 (𝑥 = 𝑀 → ⟨𝑥, 𝐾⟩ = ⟨𝑀, 𝐾⟩)
1817breq1d 5115 . . . . . 6 (𝑥 = 𝑀 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑀, 𝐾⟩ PolyAP 𝑓))
1918orbi1d 915 . . . . 5 (𝑥 = 𝑀 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
2019rexralbidv 3214 . . . 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 340 . . 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 oveq1 7364 . . . . . . . 8 (𝑠 = 𝑅 → (𝑠m (1...𝑛)) = (𝑅m (1...𝑛)))
2322raleqdv 3313 . . . . . . 7 (𝑠 = 𝑅 → (∀𝑓 ∈ (𝑠m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
2423rexbidv 3175 . . . . . 6 (𝑠 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
25 vdwlem9.s . . . . . 6 (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝐾 MonoAP 𝑓)
26 vdw.r . . . . . 6 (𝜑𝑅 ∈ Fin)
2724, 25, 26rspcdva 3582 . . . . 5 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓)
28 oveq2 7365 . . . . . . . 8 (𝑛 = 𝑤 → (1...𝑛) = (1...𝑤))
2928oveq2d 7373 . . . . . . 7 (𝑛 = 𝑤 → (𝑅m (1...𝑛)) = (𝑅m (1...𝑤)))
3029raleqdv 3313 . . . . . 6 (𝑛 = 𝑤 → (∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑓))
3130cbvrexvw 3226 . . . . 5 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑓)
3227, 31sylib 217 . . . 4 (𝜑 → ∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑓)
33 breq2 5109 . . . . . . 7 (𝑓 = 𝑔 → (𝐾 MonoAP 𝑓𝐾 MonoAP 𝑔))
3433cbvralvw 3225 . . . . . 6 (∀𝑓 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑔)
35 2nn 12226 . . . . . . . 8 2 ∈ ℕ
36 simpr 485 . . . . . . . 8 ((𝜑𝑤 ∈ ℕ) → 𝑤 ∈ ℕ)
37 nnmulcl 12177 . . . . . . . 8 ((2 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (2 · 𝑤) ∈ ℕ)
3835, 36, 37sylancr 587 . . . . . . 7 ((𝜑𝑤 ∈ ℕ) → (2 · 𝑤) ∈ ℕ)
3926adantr 481 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ℕ) → 𝑅 ∈ Fin)
40 ovex 7390 . . . . . . . . . . 11 (1...(2 · 𝑤)) ∈ V
41 elmapg 8778 . . . . . . . . . . 11 ((𝑅 ∈ Fin ∧ (1...(2 · 𝑤)) ∈ V) → (𝑓 ∈ (𝑅m (1...(2 · 𝑤))) ↔ 𝑓:(1...(2 · 𝑤))⟶𝑅))
4239, 40, 41sylancl 586 . . . . . . . . . 10 ((𝜑𝑤 ∈ ℕ) → (𝑓 ∈ (𝑅m (1...(2 · 𝑤))) ↔ 𝑓:(1...(2 · 𝑤))⟶𝑅))
4342biimpa 477 . . . . . . . . 9 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓 ∈ (𝑅m (1...(2 · 𝑤)))) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
44 simplr 767 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
45 elfznn 13470 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1...𝑤) → 𝑦 ∈ ℕ)
4645adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ∈ ℕ)
4746nnred 12168 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ∈ ℝ)
48 simpllr 774 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℕ)
4948nnred 12168 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℝ)
50 elfzle2 13445 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1...𝑤) → 𝑦𝑤)
5150adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦𝑤)
5247, 49, 49, 51leadd1dd 11769 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ≤ (𝑤 + 𝑤))
5348nncnd 12169 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℂ)
54532timesd 12396 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) = (𝑤 + 𝑤))
5552, 54breqtrrd 5133 . . . . . . . . . . . . . . 15 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ≤ (2 · 𝑤))
5646, 48nnaddcld 12205 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ ℕ)
57 nnuz 12806 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
5856, 57eleqtrdi 2848 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ (ℤ‘1))
5938ad2antrr 724 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) ∈ ℕ)
6059nnzd 12526 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) ∈ ℤ)
61 elfz5 13433 . . . . . . . . . . . . . . . 16 (((𝑦 + 𝑤) ∈ (ℤ‘1) ∧ (2 · 𝑤) ∈ ℤ) → ((𝑦 + 𝑤) ∈ (1...(2 · 𝑤)) ↔ (𝑦 + 𝑤) ≤ (2 · 𝑤)))
6258, 60, 61syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → ((𝑦 + 𝑤) ∈ (1...(2 · 𝑤)) ↔ (𝑦 + 𝑤) ≤ (2 · 𝑤)))
6355, 62mpbird 256 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ (1...(2 · 𝑤)))
6444, 63ffvelcdmd 7036 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑓‘(𝑦 + 𝑤)) ∈ 𝑅)
65 fvoveq1 7380 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑓‘(𝑥 + 𝑤)) = (𝑓‘(𝑦 + 𝑤)))
6665cbvmptv 5218 . . . . . . . . . . . . 13 (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) = (𝑦 ∈ (1...𝑤) ↦ (𝑓‘(𝑦 + 𝑤)))
6764, 66fmptd 7062 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅)
68 ovex 7390 . . . . . . . . . . . . . 14 (1...𝑤) ∈ V
69 elmapg 8778 . . . . . . . . . . . . . 14 ((𝑅 ∈ Fin ∧ (1...𝑤) ∈ V) → ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅m (1...𝑤)) ↔ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅))
7039, 68, 69sylancl 586 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ℕ) → ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅m (1...𝑤)) ↔ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅))
7170biimpar 478 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℕ) ∧ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅m (1...𝑤)))
7267, 71syldan 591 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅m (1...𝑤)))
73 breq2 5109 . . . . . . . . . . . 12 (𝑔 = (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) → (𝐾 MonoAP 𝑔𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤)))))
7473rspcv 3577 . . . . . . . . . . 11 ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅m (1...𝑤)) → (∀𝑔 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑔𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤)))))
7572, 74syl 17 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (∀𝑔 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑔𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤)))))
76 2nn0 12430 . . . . . . . . . . . . 13 2 ∈ ℕ0
77 vdwlem9.k . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ (ℤ‘2))
7877ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → 𝐾 ∈ (ℤ‘2))
79 eluznn0 12842 . . . . . . . . . . . . 13 ((2 ∈ ℕ0𝐾 ∈ (ℤ‘2)) → 𝐾 ∈ ℕ0)
8076, 78, 79sylancr 587 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → 𝐾 ∈ ℕ0)
8168, 80, 67vdwmc 16850 . . . . . . . . . . 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})))
8239ad2antrr 724 . . . . . . . . . . . . . . . 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)
8378adantr 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))
84 simpllr 774 . . . . . . . . . . . . . . . 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)
85 simplr 767 . . . . . . . . . . . . . . . 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)
86 vex 3449 . . . . . . . . . . . . . . . 16 𝑐 ∈ V
87 simprll 777 . . . . . . . . . . . . . . . 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)
88 simprlr 778 . . . . . . . . . . . . . . . 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)
89 simprr 771 . . . . . . . . . . . . . . . 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}))
9082, 83, 84, 85, 86, 87, 88, 89, 66vdwlem8 16860 . . . . . . . . . . . . . . 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)
9190orcd 871 . . . . . . . . . . . . . 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))
9291expr 457 . . . . . . . . . . . . 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)))
9392rexlimdvva 3205 . . . . . . . . . . . 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)))
9493exlimdv 1936 . . . . . . . . . . 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)))
9581, 94sylbid 239 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
9675, 95syld 47 . . . . . . . . 9 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (∀𝑔 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑔 → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
9743, 96syldan 591 . . . . . . . 8 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓 ∈ (𝑅m (1...(2 · 𝑤)))) → (∀𝑔 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑔 → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
9897ralrimdva 3151 . . . . . . 7 ((𝜑𝑤 ∈ ℕ) → (∀𝑔 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑔 → ∀𝑓 ∈ (𝑅m (1...(2 · 𝑤)))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
99 oveq2 7365 . . . . . . . . . 10 (𝑛 = (2 · 𝑤) → (1...𝑛) = (1...(2 · 𝑤)))
10099oveq2d 7373 . . . . . . . . 9 (𝑛 = (2 · 𝑤) → (𝑅m (1...𝑛)) = (𝑅m (1...(2 · 𝑤))))
101100raleqdv 3313 . . . . . . . 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)))
102101rspcev 3581 . . . . . . 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))
10338, 98, 102syl6an 682 . . . . . 6 ((𝜑𝑤 ∈ ℕ) → (∀𝑔 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑔 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
10434, 103biimtrid 241 . . . . 5 ((𝜑𝑤 ∈ ℕ) → (∀𝑓 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
105104rexlimdva 3152 . . . 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)))
10632, 105mpd 15 . . 3 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
107 breq2 5109 . . . . . . . . . 10 (𝑓 = 𝑔 → (⟨𝑚, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑔))
108 breq2 5109 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝐾 + 1) MonoAP 𝑓 ↔ (𝐾 + 1) MonoAP 𝑔))
109107, 108orbi12d 917 . . . . . . . . 9 (𝑓 = 𝑔 → ((⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)))
110109cbvralvw 3225 . . . . . . . 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))
11129raleqdv 3313 . . . . . . . 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)))
112110, 111bitrid 282 . . . . . . 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)))
113112cbvrexvw 3226 . . . . . 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))
114 oveq2 7365 . . . . . . . . . . . . 13 (𝑛 = 𝑣 → (1...𝑛) = (1...𝑣))
115114oveq2d 7373 . . . . . . . . . . . 12 (𝑛 = 𝑣 → (𝑠m (1...𝑛)) = (𝑠m (1...𝑣)))
116115raleqdv 3313 . . . . . . . . . . 11 (𝑛 = 𝑣 → (∀𝑓 ∈ (𝑠m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠m (1...𝑣))𝐾 MonoAP 𝑓))
117116cbvrexvw 3226 . . . . . . . . . 10 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑣 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑣))𝐾 MonoAP 𝑓)
118 oveq1 7364 . . . . . . . . . . . 12 (𝑠 = (𝑅m (1...𝑤)) → (𝑠m (1...𝑣)) = ((𝑅m (1...𝑤)) ↑m (1...𝑣)))
119118raleqdv 3313 . . . . . . . . . . 11 (𝑠 = (𝑅m (1...𝑤)) → (∀𝑓 ∈ (𝑠m (1...𝑣))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ ((𝑅m (1...𝑤)) ↑m (1...𝑣))𝐾 MonoAP 𝑓))
120119rexbidv 3175 . . . . . . . . . 10 (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))
121117, 120bitrid 282 . . . . . . . . 9 (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))
12225ad2antrr 724 . . . . . . . . 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)
12326ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅m (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → 𝑅 ∈ Fin)
124 fzfi 13877 . . . . . . . . . 10 (1...𝑤) ∈ Fin
125 mapfi 9292 . . . . . . . . . 10 ((𝑅 ∈ Fin ∧ (1...𝑤) ∈ Fin) → (𝑅m (1...𝑤)) ∈ Fin)
126123, 124, 125sylancl 586 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅m (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → (𝑅m (1...𝑤)) ∈ Fin)
127121, 122, 126rspcdva 3582 . . . . . . . 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)
128 simprll 777 . . . . . . . . . . 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)
129 simprrl 779 . . . . . . . . . . 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)
130 nnmulcl 12177 . . . . . . . . . . . . 13 ((2 ∈ ℕ ∧ 𝑣 ∈ ℕ) → (2 · 𝑣) ∈ ℕ)
13135, 130mpan 688 . . . . . . . . . . . 12 (𝑣 ∈ ℕ → (2 · 𝑣) ∈ ℕ)
132 nnmulcl 12177 . . . . . . . . . . . 12 ((𝑤 ∈ ℕ ∧ (2 · 𝑣) ∈ ℕ) → (𝑤 · (2 · 𝑣)) ∈ ℕ)
133131, 132sylan2 593 . . . . . . . . . . 11 ((𝑤 ∈ ℕ ∧ 𝑣 ∈ ℕ) → (𝑤 · (2 · 𝑣)) ∈ ℕ)
134128, 129, 133syl2anc 584 . . . . . . . . . 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)
135 simp1l 1197 . . . . . . . . . . . . . 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)
136135, 26syl 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)
137135, 77syl 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))
138135, 25syl 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)
139 simp1r 1198 . . . . . . . . . . . . 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)
140 simp2ll 1240 . . . . . . . . . . . . 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)
141 simp2lr 1241 . . . . . . . . . . . . . 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))
142 breq2 5109 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑘 → (⟨𝑚, 𝐾⟩ PolyAP 𝑔 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑘))
143 breq2 5109 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑘 → ((𝐾 + 1) MonoAP 𝑔 ↔ (𝐾 + 1) MonoAP 𝑘))
144142, 143orbi12d 917 . . . . . . . . . . . . . . 15 (𝑔 = 𝑘 → ((⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑘 ∨ (𝐾 + 1) MonoAP 𝑘)))
145144cbvralvw 3225 . . . . . . . . . . . . . 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))
146141, 145sylib 217 . . . . . . . . . . . . 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))
147 simp2rl 1242 . . . . . . . . . . . . 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)
148 simp2rr 1243 . . . . . . . . . . . . 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)
149 simp3 1138 . . . . . . . . . . . . . 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)))))
150 ovex 7390 . . . . . . . . . . . . . . 15 (1...(𝑤 · (2 · 𝑣))) ∈ V
151 elmapg 8778 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Fin ∧ (1...(𝑤 · (2 · 𝑣))) ∈ V) → ( ∈ (𝑅m (1...(𝑤 · (2 · 𝑣)))) ↔ :(1...(𝑤 · (2 · 𝑣)))⟶𝑅))
152136, 150, 151sylancl 586 . . . . . . . . . . . . . 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))
153149, 152mpbid 231 . . . . . . . . . . . . 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)
154 fvoveq1 7380 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣)))) = (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))))
155154cbvmptv 5218 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1...𝑤) ↦ (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))))
156 oveq1 7364 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑧 → (𝑥 − 1) = (𝑧 − 1))
157156oveq1d 7372 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → ((𝑥 − 1) + 𝑣) = ((𝑧 − 1) + 𝑣))
158157oveq2d 7373 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (𝑤 · ((𝑥 − 1) + 𝑣)) = (𝑤 · ((𝑧 − 1) + 𝑣)))
159158oveq2d 7373 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))) = (𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))
160159fveq2d 6846 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))) = (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣)))))
161160mpteq2dv 5207 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
162155, 161eqtrid 2788 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑦 ∈ (1...𝑤) ↦ (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
163162cbvmptv 5218 . . . . . . . . . . . . 13 (𝑥 ∈ (1...𝑣) ↦ (𝑦 ∈ (1...𝑤) ↦ (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣)))))) = (𝑧 ∈ (1...𝑣) ↦ (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
164136, 137, 138, 139, 140, 146, 147, 148, 153, 163vdwlem9 16861 . . . . . . . . . . . 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 ))
1651643expia 1121 . . . . . . . . . . 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 )))
166165ralrimiv 3142 . . . . . . . . . 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 ))
167 oveq2 7365 . . . . . . . . . . . . . 14 (𝑛 = (𝑤 · (2 · 𝑣)) → (1...𝑛) = (1...(𝑤 · (2 · 𝑣))))
168167oveq2d 7373 . . . . . . . . . . . . 13 (𝑛 = (𝑤 · (2 · 𝑣)) → (𝑅m (1...𝑛)) = (𝑅m (1...(𝑤 · (2 · 𝑣)))))
169168raleqdv 3313 . . . . . . . . . . . 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)))
170 breq2 5109 . . . . . . . . . . . . . 14 (𝑓 = → (⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ↔ ⟨(𝑚 + 1), 𝐾⟩ PolyAP ))
171 breq2 5109 . . . . . . . . . . . . . 14 (𝑓 = → ((𝐾 + 1) MonoAP 𝑓 ↔ (𝐾 + 1) MonoAP ))
172170, 171orbi12d 917 . . . . . . . . . . . . 13 (𝑓 = → ((⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨(𝑚 + 1), 𝐾⟩ PolyAP ∨ (𝐾 + 1) MonoAP )))
173172cbvralvw 3225 . . . . . . . . . . . 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 ))
174169, 173bitrdi 286 . . . . . . . . . . 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 )))
175174rspcev 3581 . . . . . . . . . 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))
176134, 166, 175syl2anc 584 . . . . . . . . 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))
177176anassrs 468 . . . . . . . 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))
178127, 177rexlimddv 3158 . . . . . . 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))
179178rexlimdvaa 3153 . . . . . 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)))
180113, 179biimtrid 241 . . . . 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)))
181180expcom 414 . . . 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))))
182181a2d 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))))
1836, 11, 16, 21, 106, 182nnind 12171 . 2 (𝑀 ∈ ℕ → (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
1841, 183mpcom 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 205  wa 396  wo 845  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910  {csn 4586  cop 4592   class class class wbr 5105  cmpt 5188  ccnv 5632  cima 5636  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  Fincfn 8883  1c1 11052   + caddc 11054   · cmul 11056  cle 11190  cmin 11385  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  APcvdwa 16837   MonoAP cvdwm 16838   PolyAP cvdwp 16839
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-hash 14231  df-vdwap 16840  df-vdwmc 16841  df-vdwpc 16842
This theorem is referenced by:  vdwlem11  16863
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