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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	⊢ (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 4340	. . . . . . 7 ⊢ (𝑥 = 1 → ⟨𝑥, 𝐾⟩ = ⟨1, 𝐾⟩)
3	2	breq1d 4593	. . . . . 6 ⊢ (𝑥 = 1 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨1, 𝐾⟩ PolyAP 𝑓))
4	3	orbi1d 735	. . . . 5 ⊢ (𝑥 = 1 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨1, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
5	4	rexralbidv 3040	. . . 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 329	. . 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 4340	. . . . . . 7 ⊢ (𝑥 = 𝑚 → ⟨𝑥, 𝐾⟩ = ⟨𝑚, 𝐾⟩)
8	7	breq1d 4593	. . . . . 6 ⊢ (𝑥 = 𝑚 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑚, 𝐾⟩ PolyAP 𝑓))
9	8	orbi1d 735	. . . . 5 ⊢ (𝑥 = 𝑚 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
10	9	rexralbidv 3040	. . . 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 329	. . 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 4340	. . . . . . 7 ⊢ (𝑥 = (𝑚 + 1) → ⟨𝑥, 𝐾⟩ = ⟨(𝑚 + 1), 𝐾⟩)
13	12	breq1d 4593	. . . . . 6 ⊢ (𝑥 = (𝑚 + 1) → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓))
14	13	orbi1d 735	. . . . 5 ⊢ (𝑥 = (𝑚 + 1) → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
15	14	rexralbidv 3040	. . . 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 329	. . 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 4340	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ⟨𝑥, 𝐾⟩ = ⟨𝑀, 𝐾⟩)
18	17	breq1d 4593	. . . . . 6 ⊢ (𝑥 = 𝑀 → (⟨𝑥, 𝐾⟩ PolyAP 𝑓 ↔ ⟨𝑀, 𝐾⟩ PolyAP 𝑓))
19	18	orbi1d 735	. . . . 5 ⊢ (𝑥 = 𝑀 → ((⟨𝑥, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
20	19	rexralbidv 3040	. . . 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 329	. . 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		vdw.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Fin)
23		vdwlem9.s	. . . . . 6 ⊢ (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
24		oveq1 6556	. . . . . . . . 9 ⊢ (𝑠 = 𝑅 → (𝑠 ↑_𝑚 (1...𝑛)) = (𝑅 ↑_𝑚 (1...𝑛)))
25	24	raleqdv 3121	. . . . . . . 8 ⊢ (𝑠 = 𝑅 → (∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
26	25	rexbidv 3034	. . . . . . 7 ⊢ (𝑠 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
27	26	rspcv 3278	. . . . . 6 ⊢ (𝑅 ∈ Fin → (∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
28	22, 23, 27	sylc 63	. . . . 5 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
29		oveq2 6557	. . . . . . . 8 ⊢ (𝑛 = 𝑤 → (1...𝑛) = (1...𝑤))
30	29	oveq2d 6565	. . . . . . 7 ⊢ (𝑛 = 𝑤 → (𝑅 ↑_𝑚 (1...𝑛)) = (𝑅 ↑_𝑚 (1...𝑤)))
31	30	raleqdv 3121	. . . . . 6 ⊢ (𝑛 = 𝑤 → (∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑓))
32	31	cbvrexv 3148	. . . . 5 ⊢ (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑓)
33	28, 32	sylib 207	. . . 4 ⊢ (𝜑 → ∃𝑤 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑓)
34		breq2 4587	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐾 MonoAP 𝑓 ↔ 𝐾 MonoAP 𝑔))
35	34	cbvralv 3147	. . . . . 6 ⊢ (∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))𝐾 MonoAP 𝑔)
36		2nn 11062	. . . . . . . 8 ⊢ 2 ∈ ℕ
37		simpr 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → 𝑤 ∈ ℕ)
38		nnmulcl 10920	. . . . . . . 8 ⊢ ((2 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (2 · 𝑤) ∈ ℕ)
39	36, 37, 38	sylancr 694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → (2 · 𝑤) ∈ ℕ)
40	22	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → 𝑅 ∈ Fin)
41		ovex 6577	. . . . . . . . . . 11 ⊢ (1...(2 · 𝑤)) ∈ V
42		elmapg 7757	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Fin ∧ (1...(2 · 𝑤)) ∈ V) → (𝑓 ∈ (𝑅 ↑_𝑚 (1...(2 · 𝑤))) ↔ 𝑓:(1...(2 · 𝑤))⟶𝑅))
43	40, 41, 42	sylancl 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → (𝑓 ∈ (𝑅 ↑_𝑚 (1...(2 · 𝑤))) ↔ 𝑓:(1...(2 · 𝑤))⟶𝑅))
44	43	biimpa 500	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑_𝑚 (1...(2 · 𝑤)))) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
45		simplr 788	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑓:(1...(2 · 𝑤))⟶𝑅)
46		elfznn 12241	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (1...𝑤) → 𝑦 ∈ ℕ)
47	46	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ∈ ℕ)
48	47	nnred 10912	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ∈ ℝ)
49		simpllr 795	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℕ)
50	49	nnred 10912	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℝ)
51		elfzle2 12216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (1...𝑤) → 𝑦 ≤ 𝑤)
52	51	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑦 ≤ 𝑤)
53	48, 50, 50, 52	leadd1dd 10520	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ≤ (𝑤 + 𝑤))
54	49	nncnd 10913	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → 𝑤 ∈ ℂ)
55	54	2timesd 11152	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) = (𝑤 + 𝑤))
56	53, 55	breqtrrd 4611	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ≤ (2 · 𝑤))
57	47, 49	nnaddcld 10944	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ ℕ)
58		nnuz 11599	. . . . . . . . . . . . . . . . 17 ⊢ ℕ = (ℤ_≥‘1)
59	57, 58	syl6eleq 2698	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ (ℤ_≥‘1))
60	39	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (2 · 𝑤) ∈ ℕ)
61	60	nnzd 11357	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ 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)))
63	59, 61, 62	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → ((𝑦 + 𝑤) ∈ (1...(2 · 𝑤)) ↔ (𝑦 + 𝑤) ≤ (2 · 𝑤)))
64	56, 63	mpbird 246	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑦 + 𝑤) ∈ (1...(2 · 𝑤)))
65	45, 64	ffvelrnd 6268	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) ∧ 𝑦 ∈ (1...𝑤)) → (𝑓‘(𝑦 + 𝑤)) ∈ 𝑅)
66		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑤) = (𝑦 + 𝑤))
67	66	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑓‘(𝑥 + 𝑤)) = (𝑓‘(𝑦 + 𝑤)))
68	67	cbvmptv 4678	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) = (𝑦 ∈ (1...𝑤) ↦ (𝑓‘(𝑦 + 𝑤)))
69	65, 68	fmptd 6292	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ 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))) ∈ (𝑅 ↑_𝑚 (1...𝑤)) ↔ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅))
72	40, 70, 71	sylancl 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ) → ((𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅 ↑_𝑚 (1...𝑤)) ↔ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅))
73	72	biimpar 501	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))):(1...𝑤)⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅 ↑_𝑚 (1...𝑤)))
74	69, 73	syldan 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) ∈ (𝑅 ↑_𝑚 (1...𝑤)))
75		breq2 4587	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤))) → (𝐾 MonoAP 𝑔 ↔ 𝐾 MonoAP (𝑥 ∈ (1...𝑤) ↦ (𝑓‘(𝑥 + 𝑤)))))
76	75	rspcv 3278	. . . . . . . . . . 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)))))
77	74, 76	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)))))
78		2nn0 11186	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
79		vdwlem9.k	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
80	79	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → 𝐾 ∈ (ℤ_≥‘2))
81		eluznn0 11633	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ₀ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℕ₀)
82	78, 80, 81	sylancr 694	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℕ) ∧ 𝑓:(1...(2 · 𝑤))⟶𝑅) → 𝐾 ∈ ℕ₀)
83	70, 82, 69	vdwmc 15520	. . . . . . . . . . 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})))
84	40	ad2antrr 758	. . . . . . . . . . . . . . . 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)
85	80	adantr 480	. . . . . . . . . . . . . . . 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))
86		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)
87		simplr 788	. . . . . . . . . . . . . . . 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)
88		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
89		simprll 798	. . . . . . . . . . . . . . . 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)
90		simprlr 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}))) → 𝑑 ∈ ℕ)
91		simprr 792	. . . . . . . . . . . . . . . 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}))
92	84, 85, 86, 87, 88, 89, 90, 91, 68	vdwlem8 15530	. . . . . . . . . . . . . . 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)
93	92	orcd 406	. . . . . . . . . . . . . 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))
94	93	expr 641	. . . . . . . . . . . . 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)))
95	94	rexlimdvva 3020	. . . . . . . . . . . 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)))
96	95	exlimdv 1848	. . . . . . . . . . 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)))
97	83, 96	sylbid 229	. . . . . . . . . 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)))
98	77, 97	syld 46	. . . . . . . . 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)))
99	44, 98	syldan 486	. . . . . . . 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)))
100	99	ralrimdva 2952	. . . . . . 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)))
101		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑛 = (2 · 𝑤) → (1...𝑛) = (1...(2 · 𝑤)))
102	101	oveq2d 6565	. . . . . . . . 9 ⊢ (𝑛 = (2 · 𝑤) → (𝑅 ↑_𝑚 (1...𝑛)) = (𝑅 ↑_𝑚 (1...(2 · 𝑤))))
103	102	raleqdv 3121	. . . . . . . 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)))
104	103	rspcev 3282	. . . . . . 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))
105	39, 100, 104	syl6an 566	. . . . . 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)))
106	35, 105	syl5bi 231	. . . . 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)))
107	106	rexlimdva 3013	. . . 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)))
108	33, 107	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (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))
111	109, 110	orbi12d 742	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((⟨𝑚, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)))
112	111	cbvralv 3147	. . . . . . . 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))
113	30	raleqdv 3121	. . . . . . . 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)))
114	112, 113	syl5bb 271	. . . . . . 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)))
115	114	cbvrexv 3148	. . . . . 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))
116	22	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (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) → (𝑅 ↑_𝑚 (1...𝑤)) ∈ Fin)
119	116, 117, 118	sylancl 693	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑤 ∈ ℕ ∧ ∀𝑔 ∈ (𝑅 ↑_𝑚 (1...𝑤))(⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))) → (𝑅 ↑_𝑚 (1...𝑤)) ∈ Fin)
120	23	ad2antrr 758	. . . . . . . . 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)
121		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑣 → (1...𝑛) = (1...𝑣))
122	121	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑣 → (𝑠 ↑_𝑚 (1...𝑛)) = (𝑠 ↑_𝑚 (1...𝑣)))
123	122	raleqdv 3121	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑣 → (∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
124	123	cbvrexv 3148	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑣 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓)
125		oveq1 6556	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑅 ↑_𝑚 (1...𝑤)) → (𝑠 ↑_𝑚 (1...𝑣)) = ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣)))
126	125	raleqdv 3121	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑅 ↑_𝑚 (1...𝑤)) → (∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
127	126	rexbidv 3034	. . . . . . . . . . 11 ⊢ (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))
128	124, 127	syl5bb 271	. . . . . . . . . 10 ⊢ (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))
129	128	rspcv 3278	. . . . . . . . 9 ⊢ ((𝑅 ↑_𝑚 (1...𝑤)) ∈ Fin → (∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓 → ∃𝑣 ∈ ℕ ∀𝑓 ∈ ((𝑅 ↑_𝑚 (1...𝑤)) ↑_𝑚 (1...𝑣))𝐾 MonoAP 𝑓))
130	119, 120, 129	sylc 63	. . . . . . . 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)
131		simprll 798	. . . . . . . . . . 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)
132		simprrl 800	. . . . . . . . . . 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)
133		nnmulcl 10920	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ ∧ 𝑣 ∈ ℕ) → (2 · 𝑣) ∈ ℕ)
134	36, 133	mpan 702	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ℕ → (2 · 𝑣) ∈ ℕ)
135		nnmulcl 10920	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℕ ∧ (2 · 𝑣) ∈ ℕ) → (𝑤 · (2 · 𝑣)) ∈ ℕ)
136	134, 135	sylan2 490	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℕ ∧ 𝑣 ∈ ℕ) → (𝑤 · (2 · 𝑣)) ∈ ℕ)
137	131, 132, 136	syl2anc 691	. . . . . . . . . 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)
138		simp1l 1078	. . . . . . . . . . . . . 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)
139	138, 22	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)
140	138, 79	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))
141	138, 23	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)
142		simp1r 1079	. . . . . . . . . . . . 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)
143		simp2ll 1121	. . . . . . . . . . . . 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)
144		simp2lr 1122	. . . . . . . . . . . . . 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))
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))
147	145, 146	orbi12d 742	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑘 → ((⟨𝑚, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) ↔ (⟨𝑚, 𝐾⟩ PolyAP 𝑘 ∨ (𝐾 + 1) MonoAP 𝑘)))
148	147	cbvralv 3147	. . . . . . . . . . . . . 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))
149	144, 148	sylib 207	. . . . . . . . . . . . 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))
150		simp2rl 1123	. . . . . . . . . . . . 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)
151		simp2rr 1124	. . . . . . . . . . . . 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)
152		simp3 1056	. . . . . . . . . . . . . 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)))))
153		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ (1...(𝑤 · (2 · 𝑣))) ∈ V
154		elmapg 7757	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Fin ∧ (1...(𝑤 · (2 · 𝑣))) ∈ V) → (ℎ ∈ (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣)))) ↔ ℎ:(1...(𝑤 · (2 · 𝑣)))⟶𝑅))
155	139, 153, 154	sylancl 693	. . . . . . . . . . . . . 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))
156	152, 155	mpbid 221	. . . . . . . . . . . . 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)
157		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑢 → (𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))) = (𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))))
158	157	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (ℎ‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣)))) = (ℎ‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))))
159	158	cbvmptv 4678	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (1...𝑤) ↦ (ℎ‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (ℎ‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))))
160		oveq1 6556	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → (𝑥 − 1) = (𝑧 − 1))
161	160	oveq1d 6564	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑧 → ((𝑥 − 1) + 𝑣) = ((𝑧 − 1) + 𝑣))
162	161	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → (𝑤 · ((𝑥 − 1) + 𝑣)) = (𝑤 · ((𝑧 − 1) + 𝑣)))
163	162	oveq2d 6565	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))) = (𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))
164	163	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (ℎ‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣)))) = (ℎ‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣)))))
165	164	mpteq2dv 4673	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝑢 ∈ (1...𝑤) ↦ (ℎ‘(𝑢 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (ℎ‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
166	159, 165	syl5eq 2656	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ (1...𝑤) ↦ (ℎ‘(𝑦 + (𝑤 · ((𝑥 − 1) + 𝑣))))) = (𝑢 ∈ (1...𝑤) ↦ (ℎ‘(𝑢 + (𝑤 · ((𝑧 − 1) + 𝑣))))))
167	166	cbvmptv 4678	. . . . . . . . . . . . 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))))))
168	139, 140, 141, 142, 143, 149, 150, 151, 156, 167	vdwlem9 15531	. . . . . . . . . . . 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))
169	168	3expia 1259	. . . . . . . . . . 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)))
170	169	ralrimiv 2948	. . . . . . . . . 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))
171		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑤 · (2 · 𝑣)) → (1...𝑛) = (1...(𝑤 · (2 · 𝑣))))
172	171	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑤 · (2 · 𝑣)) → (𝑅 ↑_𝑚 (1...𝑛)) = (𝑅 ↑_𝑚 (1...(𝑤 · (2 · 𝑣)))))
173	172	raleqdv 3121	. . . . . . . . . . . 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)))
174		breq2 4587	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ℎ → (⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ↔ ⟨(𝑚 + 1), 𝐾⟩ PolyAP ℎ))
175		breq2 4587	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ℎ → ((𝐾 + 1) MonoAP 𝑓 ↔ (𝐾 + 1) MonoAP ℎ))
176	174, 175	orbi12d 742	. . . . . . . . . . . . 13 ⊢ (𝑓 = ℎ → ((⟨(𝑚 + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓) ↔ (⟨(𝑚 + 1), 𝐾⟩ PolyAP ℎ ∨ (𝐾 + 1) MonoAP ℎ)))
177	176	cbvralv 3147	. . . . . . . . . . . 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))
178	173, 177	syl6bb 275	. . . . . . . . . . 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)))
179	178	rspcev 3282	. . . . . . . . . 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))
180	137, 170, 179	syl2anc 691	. . . . . . . . 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))
181	180	anassrs 678	. . . . . . . 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))
182	130, 181	rexlimddv 3017	. . . . . . 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))
183	182	rexlimdvaa 3014	. . . . . 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)))
184	115, 183	syl5bi 231	. . . . 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)))
185	184	expcom 450	. . . 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))))
186	185	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))))
187	6, 11, 16, 21, 108, 186	nnind 10915	. 2 ⊢ (𝑀 ∈ ℕ → (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨𝑀, 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
188	1, 187	mpcom 37	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 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

W3C validator