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Theorem vdwlem10 16065

Description: Lemma for vdw 16069. Set up secondary induction on 𝑀. (Contributed by Mario Carneiro, 18-Aug-2014.)

**Hypotheses**
Ref	Expression
vdw.r	⊢ (𝜑 → 𝑅 ∈ Fin)
vdwlem9.k	⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
vdwlem9.s	⊢ (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
vdwlem10.m	⊢ (𝜑 → 𝑀 ∈ ℕ)

**Assertion**
Ref	Expression
vdwlem10	⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (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 ℎ 𝑘 𝑚 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdwlem10.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2		opeq1 4623	. . . . . . 7 ⊢ (𝑥 = 1 → ⟨𝑥, 𝐾⟩ = ⟨1, 𝐾⟩)
3	2	breq1d 4883	. . . . . 6 ⊢ (𝑥 = 1 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨1, 𝐾⟩ PolyAP 𝑓))
4	3	orbi1d 947	. . . . 5 ⊢ (𝑥 = 1 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
5	4	rexralbidv 3268	. . . 4 ⊢ (𝑥 = 1 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
6	5	imbi2d 332	. . 3 ⊢ (𝑥 = 1 → ((𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) ↔ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))))
7		opeq1 4623	. . . . . . 7 ⊢ (𝑥 = 𝑚 → ⟨𝑥, 𝐾⟩ = ⟨𝑚, 𝐾⟩)
8	7	breq1d 4883	. . . . . 6 ⊢ (𝑥 = 𝑚 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑓))
9	8	orbi1d 947	. . . . 5 ⊢ (𝑥 = 𝑚 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
10	9	rexralbidv 3268	. . . 4 ⊢ (𝑥 = 𝑚 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
11	10	imbi2d 332	. . 3 ⊢ (𝑥 = 𝑚 → ((𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) ↔ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))))
12		opeq1 4623	. . . . . . 7 ⊢ (𝑥 = (𝑚 + 1) → ⟨𝑥, 𝐾⟩ = ⟨(𝑚 + 1), 𝐾⟩)
13	12	breq1d 4883	. . . . . 6 ⊢ (𝑥 = (𝑚 + 1) → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓))
14	13	orbi1d 947	. . . . 5 ⊢ (𝑥 = (𝑚 + 1) → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
15	14	rexralbidv 3268	. . . 4 ⊢ (𝑥 = (𝑚 + 1) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
16	15	imbi2d 332	. . 3 ⊢ (𝑥 = (𝑚 + 1) → ((𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) ↔ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))))
17		opeq1 4623	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ⟨𝑥, 𝐾⟩ = ⟨𝑀, 𝐾⟩)
18	17	breq1d 4883	. . . . . 6 ⊢ (𝑥 = 𝑀 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑀, 𝐾⟩ PolyAP 𝑓))
19	18	orbi1d 947	. . . . 5 ⊢ (𝑥 = 𝑀 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
20	19	rexralbidv 3268	. . . 4 ⊢ (𝑥 = 𝑀 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
21	20	imbi2d 332	. . 3 ⊢ (𝑥 = 𝑀 → ((𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) ↔ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))))
22		oveq1 6912	. . . . . . . 8 ⊢ (𝑠 = 𝑅 → (𝑠 ↑_𝑚 (1...𝑛)) = (𝑅 ↑_𝑚 (1...𝑛)))
23	22	raleqdv 3356	. . . . . . 7 ⊢ (𝑠 = 𝑅 → (∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
24	23	rexbidv 3262	. . . . . 6 ⊢ (𝑠 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
25		vdwlem9.s	. . . . . 6 ⊢ (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
26		vdw.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Fin)
27	24, 25, 26	rspcdva 3532	. . . . 5 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
28		oveq2 6913	. . . . . . . 8 ⊢ (𝑛 = 𝑤 → (1...𝑛) = (1...𝑤))
29	28	oveq2d 6921	. . . . . . 7 ⊢ (𝑛 = 𝑤 → (𝑅 ↑_𝑚 (1...𝑛)) = (𝑅 ↑_𝑚 (1...𝑤)))
30	29	raleqdv 3356	. . . . . 6 ⊢ (𝑛 = 𝑤 → (∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑓))
31	30	cbvrexv 3384	. . . . 5 ⊢ (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑓)
32	27, 31	sylib 210	. . . 4 ⊢ (𝜑 → ∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑓)
33		breq2 4877	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐾 MonoAP 𝑓 ↔ 𝐾 MonoAP 𝑔))
34	33	cbvralv 3383	. . . . . 6 ⊢ (∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑔)
35		2nn 11424	. . . . . . . 8 ⊢ 2 ∈ ℕ
36		simpr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → 𝑤 ∈ ℕ)
37		nnmulcl 11375	. . . . . . . 8 ⊢ ((2 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (2 · 𝑤) ∈ ℕ)
38	35, 36, 37	sylancr 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → (2 · 𝑤) ∈ ℕ)
39	26	adantr 474	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → 𝑅 ∈ Fin)
40		ovex 6937	. . . . . . . . . . 11 ⊢ (1...(2 · 𝑤)) ∈ V
41		elmapg 8135	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Fin ∧ (1...(2 · 𝑤)) ∈ V) → (𝑓 ∈ (𝑅 ↑_𝑚 (1...(2 · 𝑤))) ↔ 𝑓:(1...(2 · 𝑤))⟶𝑅))
42	39, 40, 41	sylancl 582	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → (𝑓 ∈ (𝑅 ↑_𝑚 (1...(2 · 𝑤))) ↔ 𝑓:(1...(2 · 𝑤))⟶𝑅))
43	42	biimpa 470	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑_𝑚 (1...(2 · 𝑤)))) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
44		simplr 787	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
45		elfznn 12663	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (1...𝑤) → 𝑦 ∈ ℕ)
46	45	adantl 475	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ∈ ℕ)
47	46	nnred 11367	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ∈ ℝ)
48		simpllr 795	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℕ)
49	48	nnred 11367	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℝ)
50		elfzle2 12638	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (1...𝑤) → 𝑦 ≤ 𝑤)
51	50	adantl 475	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ≤ 𝑤)
52	47, 49, 49, 51	leadd1dd 10966	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ≤ (𝑤 + 𝑤))
53	48	nncnd 11368	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℂ)
54	53	2timesd 11601	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) = (𝑤 + 𝑤))
55	52, 54	breqtrrd 4901	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ≤ (2 · 𝑤))
56	46, 48	nnaddcld 11403	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ ℕ)
57		nnuz 12005	. . . . . . . . . . . . . . . . 17 ⊢ ℕ = (ℤ_≥‘1)
58	56, 57	syl6eleq 2916	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ (ℤ_≥‘1))
59	38	ad2antrr 719	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) ∈ ℕ)
60	59	nnzd 11809	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) ∈ ℤ)
61		elfz5 12627	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 + 𝑤) ∈ (ℤ_≥‘1) ∧ (2 · 𝑤) ∈ ℤ) → ((𝑦 + 𝑤) ∈ (1...(2 · 𝑤)) ↔ (𝑦 + 𝑤) ≤ (2 · 𝑤)))
62	58, 60, 61	syl2anc 581	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → ((𝑦 + 𝑤) ∈ (1...(2 · 𝑤)) ↔ (𝑦 + 𝑤) ≤ (2 · 𝑤)))
63	55, 62	mpbird 249	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ (1...(2 · 𝑤)))
64	44, 63	ffvelrnd 6609	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑓‘(𝑦 + 𝑤)) ∈ 𝑅)
65		fvoveq1 6928	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑓‘(𝑥 + 𝑤)) = (𝑓‘(𝑦 + 𝑤)))
66	65	cbvmptv 4973	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) = (𝑦 ∈ (1...𝑤) ↦ (𝑓‘(𝑦 + 𝑤)))
67	64, 66	fmptd 6633	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅)
68		ovex 6937	. . . . . . . . . . . . . 14 ⊢ (1...𝑤) ∈ V
69		elmapg 8135	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Fin ∧ (1...𝑤) ∈ V) → ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅 ↑_𝑚 (1...𝑤)) ↔ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅))
70	39, 68, 69	sylancl 582	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅 ↑_𝑚 (1...𝑤)) ↔ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅))
71	70	biimpar 471	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅 ↑_𝑚 (1...𝑤)))
72	67, 71	syldan 587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅 ↑_𝑚 (1...𝑤)))
73		breq2 4877	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) → (𝐾 MonoAP 𝑔 ↔ 𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤)))))
74	73	rspcv 3522	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅 ↑_𝑚 (1...𝑤)) → (∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑔 → 𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤)))))
75	72, 74	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑔 → 𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤)))))
76		2nn0 11637	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
77		vdwlem9.k	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
78	77	ad2antrr 719	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → 𝐾 ∈ (ℤ_≥‘2))
79		eluznn0 12040	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ₀ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℕ₀)
80	76, 78, 79	sylancr 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → 𝐾 ∈ ℕ₀)
81	68, 80, 67	vdwmc 16053	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡(𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐})))
82	39	ad2antrr 719	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡(𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}))) → 𝑅 ∈ Fin)
83	78	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ 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 795	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ 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 787	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ 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 3417	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
87		simprll 799	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ 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 800	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ 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 791	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ 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}))
90	82, 83, 84, 85, 86, 87, 88, 89, 66	vdwlem8 16063	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ 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)
91	90	orcd 906	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ 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))
92	91	expr 450	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ 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)))
93	92	rexlimdvva 3248	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ 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)))
94	93	exlimdv 2034	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡(𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) “ {𝑐}) → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
95	81, 94	sylbid 232	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
96	75, 95	syld 47	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑔 → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
97	43, 96	syldan 587	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑_𝑚 (1...(2 · 𝑤)))) → (∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑔 → (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
98	97	ralrimdva 3178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → (∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑔 → ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...(2 · 𝑤)))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
99		oveq2 6913	. . . . . . . . . 10 ⊢ (𝑛 = (2 · 𝑤) → (1...𝑛) = (1...(2 · 𝑤)))
100	99	oveq2d 6921	. . . . . . . . 9 ⊢ (𝑛 = (2 · 𝑤) → (𝑅 ↑_𝑚 (1...𝑛)) = (𝑅 ↑_𝑚 (1...(2 · 𝑤))))
101	100	raleqdv 3356	. . . . . . . 8 ⊢ (𝑛 = (2 · 𝑤) → (∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...(2 · 𝑤)))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
102	101	rspcev 3526	. . . . . . 7 ⊢ (((2 · 𝑤) ∈ ℕ ∧ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...(2 · 𝑤)))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
103	38, 98, 102	syl6an 676	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → (∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑔 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
104	34, 103	syl5bi 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → (∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
105	104	rexlimdva 3240	. . . 4 ⊢ (𝜑 → (∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
106	32, 105	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
107		breq2 4877	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (⟨𝑚, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑔))
108		breq2 4877	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((𝐾 + 1) MonoAP 𝑓 ↔ (𝐾 + 1) MonoAP 𝑔))
109	107, 108	orbi12d 949	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)))
110	109	cbvralv 3383	. . . . . . . 8 ⊢ (∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
111	29	raleqdv 3356	. . . . . . . 8 ⊢ (𝑛 = 𝑤 → (∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) ↔ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)))
112	110, 111	syl5bb 275	. . . . . . 7 ⊢ (𝑛 = 𝑤 → (∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)))
113	112	cbvrexv 3384	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∃𝑤 ∈ ℕ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
114		oveq2 6913	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑣 → (1...𝑛) = (1...𝑣))
115	114	oveq2d 6921	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑣 → (𝑠 ↑_𝑚 (1...𝑛)) = (𝑠 ↑_𝑚 (1...𝑣)))
116	115	raleqdv 3356	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑣 → (∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
117	116	cbvrexv 3384	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑣 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)
118		oveq1 6912	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑅 ↑_𝑚 (1...𝑤)) → (𝑠 ↑_𝑚 (1...𝑣)) = ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣)))
119	118	raleqdv 3356	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑅 ↑_𝑚 (1...𝑤)) → (∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
120	119	rexbidv 3262	. . . . . . . . . 10 ⊢ (𝑠 = (𝑅 ↑_𝑚 (1...𝑤)) → (∃𝑣 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓 ↔ ∃𝑣 ∈ ℕ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
121	117, 120	syl5bb 275	. . . . . . . . 9 ⊢ (𝑠 = (𝑅 ↑_𝑚 (1...𝑤)) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑣 ∈ ℕ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
122	25	ad2antrr 719	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
123	26	ad2antrr 719	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → 𝑅 ∈ Fin)
124		fzfi 13066	. . . . . . . . . 10 ⊢ (1...𝑤) ∈ Fin
125		mapfi 8531	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Fin ∧ (1...𝑤) ∈ Fin) → (𝑅 ↑_𝑚 (1...𝑤)) ∈ Fin)
126	123, 124, 125	sylancl 582	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → (𝑅 ↑_𝑚 (1...𝑤)) ∈ Fin)
127	121, 122, 126	rspcdva 3532	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → ∃𝑣 ∈ ℕ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)
128		simprll 799	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))) → 𝑤 ∈ ℕ)
129		simprrl 801	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))) → 𝑣 ∈ ℕ)
130		nnmulcl 11375	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ ∧ 𝑣 ∈ ℕ) → (2 · 𝑣) ∈ ℕ)
131	35, 130	mpan 683	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ℕ → (2 · 𝑣) ∈ ℕ)
132		nnmulcl 11375	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ ∧ (2 · 𝑣) ∈ ℕ) → (𝑤 · (2 · 𝑣)) ∈ ℕ)
133	131, 132	sylan2 588	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ ∧ 𝑣 ∈ ℕ) → (𝑤 · (2 · 𝑣)) ∈ ℕ)
134	128, 129, 133	syl2anc 581	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))) → (𝑤 · (2 · 𝑣)) ∈ ℕ)
135		simp1l 1260	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → 𝜑)
136	135, 26	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → 𝑅 ∈ Fin)
137	135, 77	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → 𝐾 ∈ (ℤ_≥‘2))
138	135, 25	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
139		simp1r 1261	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → 𝑚 ∈ ℕ)
140		simp2ll 1327	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → 𝑤 ∈ ℕ)
141		simp2lr 1328	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
142		breq2 4877	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑘 → (⟨𝑚, 𝐾⟩ PolyAP 𝑔 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑘))
143		breq2 4877	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑘 → ((𝐾 + 1) MonoAP 𝑔 ↔ (𝐾 + 1) MonoAP 𝑘))
144	142, 143	orbi12d 949	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑘 → ((⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑘 ∨ (𝐾 + 1) MonoAP 𝑘)))
145	144	cbvralv 3383	. . . . . . . . . . . . . 14 ⊢ (∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) ↔ ∀𝑘 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑘 ∨ (𝐾 + 1) MonoAP 𝑘))
146	141, 145	sylib 210	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → ∀𝑘 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑘 ∨ (𝐾 + 1) MonoAP 𝑘))
147		simp2rl 1329	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → 𝑣 ∈ ℕ)
148		simp2rr 1330	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)
149		simp3 1174	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣)))))
150		ovex 6937	. . . . . . . . . . . . . . 15 ⊢ (1...(𝑤 · (2 · 𝑣))) ∈ V
151		elmapg 8135	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Fin ∧ (1...(𝑤 · (2 · 𝑣))) ∈ V) → (ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣)))) ↔ ℎ:(1...(𝑤 · (2 · 𝑣)))⟶𝑅))
152	136, 150, 151	sylancl 582	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → (ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣)))) ↔ ℎ:(1...(𝑤 · (2 · 𝑣)))⟶𝑅))
153	149, 152	mpbid 224	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → ℎ:(1...(𝑤 · (2 · 𝑣)))⟶𝑅)
154		fvoveq1 6928	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (ℎ‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣)))) = (ℎ‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))))
155	154	cbvmptv 4973	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (1...𝑤) ↦ (ℎ‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (ℎ‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))))
156		oveq1 6912	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → (𝑥 − 1) = (𝑧 − 1))
157	156	oveq1d 6920	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑧 → ((𝑥 − 1) + 𝑣) = ((𝑧 − 1) + 𝑣))
158	157	oveq2d 6921	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → (𝑤 · ((𝑥 − 1) + 𝑣)) = (𝑤 · ((𝑧 − 1) + 𝑣)))
159	158	oveq2d 6921	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))) = (𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))
160	159	fveq2d 6437	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (ℎ‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))) = (ℎ‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣)))))
161	160	mpteq2dv 4968	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝑢 ∈ (1...𝑤) ↦ (ℎ‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (ℎ‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
162	155, 161	syl5eq 2873	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ (1...𝑤) ↦ (ℎ‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (ℎ‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
163	162	cbvmptv 4973	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...𝑣) ↦ (𝑦 ∈ (1...𝑤) ↦ (ℎ‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣)))))) = (𝑧 ∈ (1...𝑣) ↦ (𝑢 ∈ (1...𝑤) ↦ (ℎ‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
164	136, 137, 138, 139, 140, 146, 147, 148, 153, 163	vdwlem9 16064	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) ∧ ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))) → (⟨(𝑚 + 1), 𝐾⟩ PolyAP ℎ ∨ (𝐾 + 1) MonoAP ℎ))
165	164	3expia 1156	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))) → (ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣)))) → (⟨(𝑚 + 1), 𝐾⟩ PolyAP ℎ ∨ (𝐾 + 1) MonoAP ℎ)))
166	165	ralrimiv 3174	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))) → ∀ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))(⟨(𝑚 + 1), 𝐾⟩ PolyAP ℎ ∨ (𝐾 + 1) MonoAP ℎ))
167		oveq2 6913	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑤 · (2 · 𝑣)) → (1...𝑛) = (1...(𝑤 · (2 · 𝑣))))
168	167	oveq2d 6921	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑤 · (2 · 𝑣)) → (𝑅 ↑_𝑚 (1...𝑛)) = (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣)))))
169	168	raleqdv 3356	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑤 · (2 · 𝑣)) → (∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
170		breq2 4877	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ℎ → (⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ↔ ⟨(𝑚 + 1), 𝐾⟩ PolyAP ℎ))
171		breq2 4877	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ℎ → ((𝐾 + 1) MonoAP 𝑓 ↔ (𝐾 + 1) MonoAP ℎ))
172	170, 171	orbi12d 949	. . . . . . . . . . . . 13 ⊢ (𝑓 = ℎ → ((⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨(𝑚 + 1), 𝐾⟩ PolyAP ℎ ∨ (𝐾 + 1) MonoAP ℎ)))
173	172	cbvralv 3383	. . . . . . . . . . . 12 ⊢ (∀𝑓 ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∀ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))(⟨(𝑚 + 1), 𝐾⟩ PolyAP ℎ ∨ (𝐾 + 1) MonoAP ℎ))
174	169, 173	syl6bb 279	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑤 · (2 · 𝑣)) → (∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ ∀ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))(⟨(𝑚 + 1), 𝐾⟩ PolyAP ℎ ∨ (𝐾 + 1) MonoAP ℎ)))
175	174	rspcev 3526	. . . . . . . . . 10 ⊢ (((𝑤 · (2 · 𝑣)) ∈ ℕ ∧ ∀ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣))))(⟨(𝑚 + 1), 𝐾⟩ PolyAP ℎ ∨ (𝐾 + 1) MonoAP ℎ)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
176	134, 166, 175	syl2anc 581	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ((𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
177	176	anassrs 461	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) ∧ (𝑣 ∈ ℕ ∧ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
178	127, 177	rexlimddv 3245	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
179	178	rexlimdvaa 3241	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∃𝑤 ∈ ℕ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
180	113, 179	syl5bi 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
181	180	expcom 404	. . . 4 ⊢ (𝑚 ∈ ℕ → (𝜑 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))))
182	181	a2d 29	. . 3 ⊢ (𝑚 ∈ ℕ → ((𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)) → (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))))
183	6, 11, 16, 21, 106, 182	nnind 11370	. 2 ⊢ (𝑀 ∈ ℕ → (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
184	1, 183	mpcom 38	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 880 ∧ w3a 1113 = wceq 1658 ∃wex 1880 ∈ wcel 2166 ∀wral 3117 ∃wrex 3118 Vcvv 3414 ⊆ wss 3798 {csn 4397 ⟨cop 4403 class class class wbr 4873 ↦ cmpt 4952 ^◡ccnv 5341 “ cima 5345 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ↑_𝑚 cmap 8122 Fincfn 8222 1c1 10253 + caddc 10255 · cmul 10257 ≤ cle 10392 − cmin 10585 ℕcn 11350 2c2 11406 ℕ₀cn0 11618 ℤcz 11704 ℤ_≥cuz 11968 ...cfz 12619 APcvdwa 16040 MonoAP cvdwm 16041 PolyAP cvdwp 16042

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-fz 12620 df-hash 13411 df-vdwap 16043 df-vdwmc 16044 df-vdwpc 16045

This theorem is referenced by: vdwlem11 16066

W3C validator