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Theorem vdwlem10 17023
Description: Lemma for vdw 17027. 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 4877 . . . . . . 7 (𝑥 = 1 → ⟨𝑥, 𝐾⟩ = ⟨1, 𝐾⟩)
32breq1d 5157 . . . . . 6 (𝑥 = 1 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨1, 𝐾⟩ PolyAP 𝑓))
43orbi1d 916 . . . . 5 (𝑥 = 1 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
54rexralbidv 3220 . . . 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 4877 . . . . . . 7 (𝑥 = 𝑚 → ⟨𝑥, 𝐾⟩ = ⟨𝑚, 𝐾⟩)
87breq1d 5157 . . . . . 6 (𝑥 = 𝑚 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑓))
98orbi1d 916 . . . . 5 (𝑥 = 𝑚 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
109rexralbidv 3220 . . . 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 4877 . . . . . . 7 (𝑥 = (𝑚 + 1) → ⟨𝑥, 𝐾⟩ = ⟨(𝑚 + 1), 𝐾⟩)
1312breq1d 5157 . . . . . 6 (𝑥 = (𝑚 + 1) → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓))
1413orbi1d 916 . . . . 5 (𝑥 = (𝑚 + 1) → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
1514rexralbidv 3220 . . . 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 4877 . . . . . . 7 (𝑥 = 𝑀 → ⟨𝑥, 𝐾⟩ = ⟨𝑀, 𝐾⟩)
1817breq1d 5157 . . . . . 6 (𝑥 = 𝑀 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑀, 𝐾⟩ PolyAP 𝑓))
1918orbi1d 916 . . . . 5 (𝑥 = 𝑀 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
2019rexralbidv 3220 . . . 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 7437 . . . . . . . 8 (𝑠 = 𝑅 → (𝑠m (1...𝑛)) = (𝑅m (1...𝑛)))
2322raleqdv 3323 . . . . . . 7 (𝑠 = 𝑅 → (∀𝑓 ∈ (𝑠m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
2423rexbidv 3176 . . . . . 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 3622 . . . . 5 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓)
28 oveq2 7438 . . . . . . . 8 (𝑛 = 𝑤 → (1...𝑛) = (1...𝑤))
2928oveq2d 7446 . . . . . . 7 (𝑛 = 𝑤 → (𝑅m (1...𝑛)) = (𝑅m (1...𝑤)))
3029raleqdv 3323 . . . . . 6 (𝑛 = 𝑤 → (∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑓))
3130cbvrexvw 3235 . . . . 5 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑓)
3227, 31sylib 218 . . . 4 (𝜑 → ∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑓)
33 breq2 5151 . . . . . . 7 (𝑓 = 𝑔 → (𝐾 MonoAP 𝑓𝐾 MonoAP 𝑔))
3433cbvralvw 3234 . . . . . 6 (∀𝑓 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑅m (1...𝑤))𝐾 MonoAP 𝑔)
35 2nn 12336 . . . . . . . 8 2 ∈ ℕ
36 simpr 484 . . . . . . . 8 ((𝜑𝑤 ∈ ℕ) → 𝑤 ∈ ℕ)
37 nnmulcl 12287 . . . . . . . 8 ((2 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (2 · 𝑤) ∈ ℕ)
3835, 36, 37sylancr 587 . . . . . . 7 ((𝜑𝑤 ∈ ℕ) → (2 · 𝑤) ∈ ℕ)
3926adantr 480 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ℕ) → 𝑅 ∈ Fin)
40 ovex 7463 . . . . . . . . . . 11 (1...(2 · 𝑤)) ∈ V
41 elmapg 8877 . . . . . . . . . . 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 476 . . . . . . . . 9 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓 ∈ (𝑅m (1...(2 · 𝑤)))) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
44 simplr 769 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
45 elfznn 13589 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (1...𝑤) → 𝑦 ∈ ℕ)
4645adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ∈ ℕ)
4746nnred 12278 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ∈ ℝ)
48 simpllr 776 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℕ)
4948nnred 12278 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℝ)
50 elfzle2 13564 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1...𝑤) → 𝑦𝑤)
5150adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦𝑤)
5247, 49, 49, 51leadd1dd 11874 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ≤ (𝑤 + 𝑤))
5348nncnd 12279 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℂ)
54532timesd 12506 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) = (𝑤 + 𝑤))
5552, 54breqtrrd 5175 . . . . . . . . . . . . . . 15 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ≤ (2 · 𝑤))
5646, 48nnaddcld 12315 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ ℕ)
57 nnuz 12918 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
5856, 57eleqtrdi 2848 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ (ℤ‘1))
5938ad2antrr 726 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) ∈ ℕ)
6059nnzd 12637 . . . . . . . . . . . . . . . 16 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) ∈ ℤ)
61 elfz5 13552 . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ (1...(2 · 𝑤)))
6444, 63ffvelcdmd 7104 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑓‘(𝑦 + 𝑤)) ∈ 𝑅)
65 fvoveq1 7453 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑓‘(𝑥 + 𝑤)) = (𝑓‘(𝑦 + 𝑤)))
6665cbvmptv 5260 . . . . . . . . . . . . 13 (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) = (𝑦 ∈ (1...𝑤) ↦ (𝑓‘(𝑦 + 𝑤)))
6764, 66fmptd 7133 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅)
68 ovex 7463 . . . . . . . . . . . . . 14 (1...𝑤) ∈ V
69 elmapg 8877 . . . . . . . . . . . . . 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 477 . . . . . . . . . . . 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 5151 . . . . . . . . . . . 12 (𝑔 = (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) → (𝐾 MonoAP 𝑔𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤)))))
7473rspcv 3617 . . . . . . . . . . 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 12540 . . . . . . . . . . . . 13 2 ∈ ℕ0
77 vdwlem9.k . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ (ℤ‘2))
7877ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → 𝐾 ∈ (ℤ‘2))
79 eluznn0 12956 . . . . . . . . . . . . 13 ((2 ∈ ℕ0𝐾 ∈ (ℤ‘2)) → 𝐾 ∈ ℕ0)
8076, 78, 79sylancr 587 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → 𝐾 ∈ ℕ0)
8168, 80, 67vdwmc 17011 . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . 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 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))
84 simpllr 776 . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . 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 3481 . . . . . . . . . . . . . . . 16 𝑐 ∈ V
87 simprll 779 . . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . 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 17021 . . . . . . . . . . . . . . 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 873 . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . 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 3210 . . . . . . . . . . . 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 1930 . . . . . . . . . . 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 240 . . . . . . . . . 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 7438 . . . . . . . . . 10 (𝑛 = (2 · 𝑤) → (1...𝑛) = (1...(2 · 𝑤)))
10099oveq2d 7446 . . . . . . . . 9 (𝑛 = (2 · 𝑤) → (𝑅m (1...𝑛)) = (𝑅m (1...(2 · 𝑤))))
101100raleqdv 3323 . . . . . . . 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 3621 . . . . . . 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 684 . . . . . 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 242 . . . . 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 5151 . . . . . . . . . 10 (𝑓 = 𝑔 → (⟨𝑚, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑔))
108 breq2 5151 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝐾 + 1) MonoAP 𝑓 ↔ (𝐾 + 1) MonoAP 𝑔))
109107, 108orbi12d 918 . . . . . . . . 9 (𝑓 = 𝑔 → ((⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)))
110109cbvralvw 3234 . . . . . . . 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 3323 . . . . . . . 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 283 . . . . . . 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 3235 . . . . . 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 7438 . . . . . . . . . . . . 13 (𝑛 = 𝑣 → (1...𝑛) = (1...𝑣))
115114oveq2d 7446 . . . . . . . . . . . 12 (𝑛 = 𝑣 → (𝑠m (1...𝑛)) = (𝑠m (1...𝑣)))
116115raleqdv 3323 . . . . . . . . . . 11 (𝑛 = 𝑣 → (∀𝑓 ∈ (𝑠m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠m (1...𝑣))𝐾 MonoAP 𝑓))
117116cbvrexvw 3235 . . . . . . . . . 10 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑣 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑣))𝐾 MonoAP 𝑓)
118 oveq1 7437 . . . . . . . . . . . 12 (𝑠 = (𝑅m (1...𝑤)) → (𝑠m (1...𝑣)) = ((𝑅m (1...𝑤)) ↑m (1...𝑣)))
119118raleqdv 3323 . . . . . . . . . . 11 (𝑠 = (𝑅m (1...𝑤)) → (∀𝑓 ∈ (𝑠m (1...𝑣))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ ((𝑅m (1...𝑤)) ↑m (1...𝑣))𝐾 MonoAP 𝑓))
120119rexbidv 3176 . . . . . . . . . 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 283 . . . . . . . . 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 726 . . . . . . . . 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 726 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅m (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → 𝑅 ∈ Fin)
124 fzfi 14009 . . . . . . . . . 10 (1...𝑤) ∈ Fin
125 mapfi 9385 . . . . . . . . . 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 3622 . . . . . . . 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 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))) → 𝑤 ∈ ℕ)
129 simprrl 781 . . . . . . . . . . 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 12287 . . . . . . . . . . . . 13 ((2 ∈ ℕ ∧ 𝑣 ∈ ℕ) → (2 · 𝑣) ∈ ℕ)
13135, 130mpan 690 . . . . . . . . . . . 12 (𝑣 ∈ ℕ → (2 · 𝑣) ∈ ℕ)
132 nnmulcl 12287 . . . . . . . . . . . 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 1196 . . . . . . . . . . . . . 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 1197 . . . . . . . . . . . . 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 1239 . . . . . . . . . . . . 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 1240 . . . . . . . . . . . . . 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 5151 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑘 → (⟨𝑚, 𝐾⟩ PolyAP 𝑔 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑘))
143 breq2 5151 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑘 → ((𝐾 + 1) MonoAP 𝑔 ↔ (𝐾 + 1) MonoAP 𝑘))
144142, 143orbi12d 918 . . . . . . . . . . . . . . 15 (𝑔 = 𝑘 → ((⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑘 ∨ (𝐾 + 1) MonoAP 𝑘)))
145144cbvralvw 3234 . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . 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 1241 . . . . . . . . . . . . 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 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))))) → ∀𝑓 ∈ ((𝑅m (1...𝑤)) ↑m (1...𝑣))𝐾 MonoAP 𝑓)
149 simp3 1137 . . . . . . . . . . . . . 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 7463 . . . . . . . . . . . . . . 15 (1...(𝑤 · (2 · 𝑣))) ∈ V
151 elmapg 8877 . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 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 7453 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣)))) = (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))))
155154cbvmptv 5260 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1...𝑤) ↦ (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))))
156 oveq1 7437 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑧 → (𝑥 − 1) = (𝑧 − 1))
157156oveq1d 7445 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → ((𝑥 − 1) + 𝑣) = ((𝑧 − 1) + 𝑣))
158157oveq2d 7446 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (𝑤 · ((𝑥 − 1) + 𝑣)) = (𝑤 · ((𝑧 − 1) + 𝑣)))
159158oveq2d 7446 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))) = (𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))
160159fveq2d 6910 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))) = (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣)))))
161160mpteq2dv 5249 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
162155, 161eqtrid 2786 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝑦 ∈ (1...𝑤) ↦ (‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
163162cbvmptv 5260 . . . . . . . . . . . . 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 17022 . . . . . . . . . . . 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 1120 . . . . . . . . . . 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 7438 . . . . . . . . . . . . . 14 (𝑛 = (𝑤 · (2 · 𝑣)) → (1...𝑛) = (1...(𝑤 · (2 · 𝑣))))
168167oveq2d 7446 . . . . . . . . . . . . 13 (𝑛 = (𝑤 · (2 · 𝑣)) → (𝑅m (1...𝑛)) = (𝑅m (1...(𝑤 · (2 · 𝑣)))))
169168raleqdv 3323 . . . . . . . . . . . 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 5151 . . . . . . . . . . . . . 14 (𝑓 = → (⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ↔ ⟨(𝑚 + 1), 𝐾⟩ PolyAP ))
171 breq2 5151 . . . . . . . . . . . . . 14 (𝑓 = → ((𝐾 + 1) MonoAP 𝑓 ↔ (𝐾 + 1) MonoAP ))
172170, 171orbi12d 918 . . . . . . . . . . . . 13 (𝑓 = → ((⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨(𝑚 + 1), 𝐾⟩ PolyAP ∨ (𝐾 + 1) MonoAP )))
173172cbvralvw 3234 . . . . . . . . . . . 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 287 . . . . . . . . . . 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 3621 . . . . . . . . . 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 467 . . . . . . . 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 242 . . . . 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 413 . . . 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 12281 . 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 206  wa 395  wo 847  w3a 1086   = wceq 1536  wex 1775  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  wss 3962  {csn 4630  cop 4636   class class class wbr 5147  cmpt 5230  ccnv 5687  cima 5691  wf 6558  cfv 6562  (class class class)co 7430  m cmap 8864  Fincfn 8983  1c1 11153   + caddc 11155   · cmul 11157  cle 11293  cmin 11489  cn 12263  2c2 12318  0cn0 12523  cz 12610  cuz 12875  ...cfz 13543  APcvdwa 16998   MonoAP cvdwm 16999   PolyAP cvdwp 17000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-hash 14366  df-vdwap 17001  df-vdwmc 17002  df-vdwpc 17003
This theorem is referenced by:  vdwlem11  17024
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