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Theorem vdwlem9 16954

Description: Lemma for vdw 16959. (Contributed by Mario Carneiro, 12-Sep-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)
vdwlem9.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
vdwlem9.w	⊢ (𝜑 → 𝑊 ∈ ℕ)
vdwlem9.g	⊢ (𝜑 → ∀𝑔 ∈ (𝑅 ↑_m (1...𝑊))(⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
vdwlem9.v	⊢ (𝜑 → 𝑉 ∈ ℕ)
vdwlem9.a	⊢ (𝜑 → ∀𝑓 ∈ ((𝑅 ↑_m (1...𝑊)) ↑_m (1...𝑉))𝐾 MonoAP 𝑓)
vdwlem9.h	⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
vdwlem9.f	⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))

**Assertion**
Ref	Expression
vdwlem9	⊢ (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻))

Distinct variable groups: 𝑔,𝑛,𝑥,𝑦,𝜑 𝑥,𝑓,𝑦,𝑉 𝑓,𝑊,𝑥,𝑦 𝑓,𝑔,𝐹,𝑥,𝑦 𝑓,𝑛,𝑠,𝐾,𝑔,𝑥,𝑦 𝑓,𝑀,𝑔,𝑛,𝑥,𝑦 𝑅,𝑓,𝑔,𝑛,𝑠,𝑥,𝑦 𝑔,𝐻,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑓,𝑠) 𝐹(𝑛,𝑠) 𝐻(𝑓,𝑛,𝑠) 𝑀(𝑠) 𝑉(𝑔,𝑛,𝑠) 𝑊(𝑔,𝑛,𝑠)

Proof of Theorem vdwlem9 Dummy variables 𝑎 𝑑 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5090	. . 3 ⊢ (𝑓 = 𝐹 → (𝐾 MonoAP 𝑓 ↔ 𝐾 MonoAP 𝐹))
2		vdwlem9.a	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ ((𝑅 ↑_m (1...𝑊)) ↑_m (1...𝑉))𝐾 MonoAP 𝑓)
3		vdwlem9.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ ℕ)
4		vdwlem9.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℕ)
5		vdw.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Fin)
6		vdwlem9.h	. . . . 5 ⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
7		vdwlem9.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
8	3, 4, 5, 6, 7	vdwlem4 16949	. . . 4 ⊢ (𝜑 → 𝐹:(1...𝑉)⟶(𝑅 ↑_m (1...𝑊)))
9		ovex 7394	. . . . 5 ⊢ (𝑅 ↑_m (1...𝑊)) ∈ V
10		ovex 7394	. . . . 5 ⊢ (1...𝑉) ∈ V
11	9, 10	elmap 8813	. . . 4 ⊢ (𝐹 ∈ ((𝑅 ↑_m (1...𝑊)) ↑_m (1...𝑉)) ↔ 𝐹:(1...𝑉)⟶(𝑅 ↑_m (1...𝑊)))
12	8, 11	sylibr 234	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↑_m (1...𝑊)) ↑_m (1...𝑉)))
13	1, 2, 12	rspcdva 3566	. 2 ⊢ (𝜑 → 𝐾 MonoAP 𝐹)
14		vdwlem9.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
15		eluz2nn 12832	. . . . . 6 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
16	14, 15	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ)
17	16	nnnn0d 12492	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
18	10, 17, 8	vdwmc 16943	. . 3 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑔∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔})))
19		vdwlem9.g	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑔 ∈ (𝑅 ↑_m (1...𝑊))(⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ∀𝑔 ∈ (𝑅 ↑_m (1...𝑊))(⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
21		simprr 773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))
22	16	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝐾 ∈ ℕ)
23		simprll 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑎 ∈ ℕ)
24		simprlr 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑑 ∈ ℕ)
25		vdwapid1 16940	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → 𝑎 ∈ (𝑎(AP‘𝐾)𝑑))
26	22, 23, 24, 25	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑎 ∈ (𝑎(AP‘𝐾)𝑑))
27	21, 26	sseldd 3923	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑎 ∈ (^◡𝐹 “ {𝑔}))
28	8	ffnd 6664	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn (1...𝑉))
29	28	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝐹 Fn (1...𝑉))
30		fniniseg 7007	. . . . . . . . . . . 12 ⊢ (𝐹 Fn (1...𝑉) → (𝑎 ∈ (^◡𝐹 “ {𝑔}) ↔ (𝑎 ∈ (1...𝑉) ∧ (𝐹‘𝑎) = 𝑔)))
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑎 ∈ (^◡𝐹 “ {𝑔}) ↔ (𝑎 ∈ (1...𝑉) ∧ (𝐹‘𝑎) = 𝑔)))
32	27, 31	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑎 ∈ (1...𝑉) ∧ (𝐹‘𝑎) = 𝑔))
33	32	simprd 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝐹‘𝑎) = 𝑔)
34	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝐹:(1...𝑉)⟶(𝑅 ↑_m (1...𝑊)))
35	32	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑎 ∈ (1...𝑉))
36	34, 35	ffvelcdmd 7032	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝐹‘𝑎) ∈ (𝑅 ↑_m (1...𝑊)))
37	33, 36	eqeltrrd 2838	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑔 ∈ (𝑅 ↑_m (1...𝑊)))
38		rsp 3226	. . . . . . . 8 ⊢ (∀𝑔 ∈ (𝑅 ↑_m (1...𝑊))(⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) → (𝑔 ∈ (𝑅 ↑_m (1...𝑊)) → (⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)))
39	20, 37, 38	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
40	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑉 ∈ ℕ)
41	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑊 ∈ ℕ)
42	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑅 ∈ Fin)
43	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
44		vdwlem9.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ)
45	44	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑀 ∈ ℕ)
46		ovex 7394	. . . . . . . . . . . 12 ⊢ (1...𝑊) ∈ V
47		elmapg 8780	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Fin ∧ (1...𝑊) ∈ V) → (𝑔 ∈ (𝑅 ↑_m (1...𝑊)) ↔ 𝑔:(1...𝑊)⟶𝑅))
48	42, 46, 47	sylancl 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑔 ∈ (𝑅 ↑_m (1...𝑊)) ↔ 𝑔:(1...𝑊)⟶𝑅))
49	37, 48	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑔:(1...𝑊)⟶𝑅)
50	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝐾 ∈ (ℤ_≥‘2))
51	40, 41, 42, 43, 7, 45, 49, 50, 23, 24, 21	vdwlem7 16952	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (⟨𝑀, 𝐾⟩ PolyAP 𝑔 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝑔)))
52		olc 869	. . . . . . . . . 10 ⊢ ((𝐾 + 1) MonoAP 𝑔 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝑔))
53	52	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((𝐾 + 1) MonoAP 𝑔 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝑔)))
54	51, 53	jaod 860	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝑔)))
55		oveq1 7368	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝑥 − 1) = (𝑎 − 1))
56	55	oveq1d 7376	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((𝑥 − 1) + 𝑉) = ((𝑎 − 1) + 𝑉))
57	56	oveq2d 7377	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝑊 · ((𝑥 − 1) + 𝑉)) = (𝑊 · ((𝑎 − 1) + 𝑉)))
58	57	oveq2d 7377	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))) = (𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))
59	58	fveq2d 6839	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
60	59	mpteq2dv 5180	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))))
61	46	mptex 7172	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) ∈ V
62	60, 7, 61	fvmpt 6942	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (1...𝑉) → (𝐹‘𝑎) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))))
63	35, 62	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝐹‘𝑎) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))))
64	63, 33	eqtr3d 2774	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) = 𝑔)
65	64	breq2d 5098	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((𝐾 + 1) MonoAP (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) ↔ (𝐾 + 1) MonoAP 𝑔))
66	17	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝐾 ∈ ℕ₀)
67		peano2nn0 12471	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ₀ → (𝐾 + 1) ∈ ℕ₀)
68	66, 67	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝐾 + 1) ∈ ℕ₀)
69		nnm1nn0 12472	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ℕ → (𝑎 − 1) ∈ ℕ₀)
70	23, 69	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑎 − 1) ∈ ℕ₀)
71		nn0nnaddcl 12462	. . . . . . . . . . . . 13 ⊢ (((𝑎 − 1) ∈ ℕ₀ ∧ 𝑉 ∈ ℕ) → ((𝑎 − 1) + 𝑉) ∈ ℕ)
72	70, 40, 71	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((𝑎 − 1) + 𝑉) ∈ ℕ)
73	41, 72	nnmulcld 12224	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑊 · ((𝑎 − 1) + 𝑉)) ∈ ℕ)
74	23, 40	nnaddcld 12223	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ∈ ℕ)
75	41, 74	nnmulcld 12224	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ∈ ℕ)
76	75	nnzd 12544	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ∈ ℤ)
77		2nn 12248	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
78		nnmulcl 12192	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℕ ∧ 𝑉 ∈ ℕ) → (2 · 𝑉) ∈ ℕ)
79	77, 3, 78	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 · 𝑉) ∈ ℕ)
80	4, 79	nnmulcld 12224	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℕ)
81	80	nnzd 12544	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℤ)
82	81	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ ℤ)
83	23	nnred 12183	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑎 ∈ ℝ)
84	40	nnred 12183	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑉 ∈ ℝ)
85		elfzle2 13476	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (1...𝑉) → 𝑎 ≤ 𝑉)
86	35, 85	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑎 ≤ 𝑉)
87	83, 84, 84, 86	leadd1dd 11758	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ≤ (𝑉 + 𝑉))
88	40	nncnd 12184	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑉 ∈ ℂ)
89	88	2timesd 12414	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (2 · 𝑉) = (𝑉 + 𝑉))
90	87, 89	breqtrrd 5114	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ≤ (2 · 𝑉))
91	74	nnred 12183	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ∈ ℝ)
92	79	nnred 12183	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 · 𝑉) ∈ ℝ)
93	92	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (2 · 𝑉) ∈ ℝ)
94	41	nnred 12183	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑊 ∈ ℝ)
95	41	nngt0d 12220	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 0 < 𝑊)
96		lemul2 12002	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 + 𝑉) ∈ ℝ ∧ (2 · 𝑉) ∈ ℝ ∧ (𝑊 ∈ ℝ ∧ 0 < 𝑊)) → ((𝑎 + 𝑉) ≤ (2 · 𝑉) ↔ (𝑊 · (𝑎 + 𝑉)) ≤ (𝑊 · (2 · 𝑉))))
97	91, 93, 94, 95, 96	syl112anc 1377	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((𝑎 + 𝑉) ≤ (2 · 𝑉) ↔ (𝑊 · (𝑎 + 𝑉)) ≤ (𝑊 · (2 · 𝑉))))
98	90, 97	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ≤ (𝑊 · (2 · 𝑉)))
99		eluz2 12788	. . . . . . . . . . . . 13 ⊢ ((𝑊 · (2 · 𝑉)) ∈ (ℤ_≥‘(𝑊 · (𝑎 + 𝑉))) ↔ ((𝑊 · (𝑎 + 𝑉)) ∈ ℤ ∧ (𝑊 · (2 · 𝑉)) ∈ ℤ ∧ (𝑊 · (𝑎 + 𝑉)) ≤ (𝑊 · (2 · 𝑉))))
100	76, 82, 98, 99	syl3anbrc 1345	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ (ℤ_≥‘(𝑊 · (𝑎 + 𝑉))))
101	41	nncnd 12184	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑊 ∈ ℂ)
102		1cnd 11133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 1 ∈ ℂ)
103	70	nn0cnd 12494	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑎 − 1) ∈ ℂ)
104	103, 88	addcld 11158	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((𝑎 − 1) + 𝑉) ∈ ℂ)
105	101, 102, 104	adddid 11163	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑊 · (1 + ((𝑎 − 1) + 𝑉))) = ((𝑊 · 1) + (𝑊 · ((𝑎 − 1) + 𝑉))))
106	102, 103, 88	addassd 11161	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((1 + (𝑎 − 1)) + 𝑉) = (1 + ((𝑎 − 1) + 𝑉)))
107		ax-1cn 11090	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
108	23	nncnd 12184	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → 𝑎 ∈ ℂ)
109		pncan3 11395	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (1 + (𝑎 − 1)) = 𝑎)
110	107, 108, 109	sylancr 588	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (1 + (𝑎 − 1)) = 𝑎)
111	110	oveq1d 7376	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((1 + (𝑎 − 1)) + 𝑉) = (𝑎 + 𝑉))
112	106, 111	eqtr3d 2774	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (1 + ((𝑎 − 1) + 𝑉)) = (𝑎 + 𝑉))
113	112	oveq2d 7377	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑊 · (1 + ((𝑎 − 1) + 𝑉))) = (𝑊 · (𝑎 + 𝑉)))
114	101	mulridd 11156	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑊 · 1) = 𝑊)
115	114	oveq1d 7376	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((𝑊 · 1) + (𝑊 · ((𝑎 − 1) + 𝑉))) = (𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉))))
116	105, 113, 115	3eqtr3d 2780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) = (𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉))))
117	116	fveq2d 6839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (ℤ_≥‘(𝑊 · (𝑎 + 𝑉))) = (ℤ_≥‘(𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
118	100, 117	eleqtrd 2839	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ (ℤ_≥‘(𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
119		fvoveq1 7384	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉)))) = (𝐻‘(𝑧 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
120	119	cbvmptv 5190	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) = (𝑧 ∈ (1...𝑊) ↦ (𝐻‘(𝑧 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
121	42, 68, 41, 73, 43, 118, 120	vdwlem2 16947	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((𝐾 + 1) MonoAP (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) → (𝐾 + 1) MonoAP 𝐻))
122	65, 121	sylbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((𝐾 + 1) MonoAP 𝑔 → (𝐾 + 1) MonoAP 𝐻))
123	122	orim2d 969	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝑔) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
124	54, 123	syld 47	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → ((⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
125	39, 124	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}))) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻))
126	125	expr 456	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
127	126	rexlimdvva 3195	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
128	127	exlimdv 1935	. . 3 ⊢ (𝜑 → (∃𝑔∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (^◡𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
129	18, 128	sylbid 240	. 2 ⊢ (𝜑 → (𝐾 MonoAP 𝐹 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
130	13, 129	mpd 15	1 ⊢ (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 Vcvv 3430 ⊆ wss 3890 {csn 4568 ⟨cop 4574 class class class wbr 5086 ↦ cmpt 5167 ^◡ccnv 5624 “ cima 5628 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 ↑_m cmap 8767 Fincfn 8887 ℂcc 11030 ℝcr 11031 0cc0 11032 1c1 11033 + caddc 11035 · cmul 11037 < clt 11173 ≤ cle 11174 − cmin 11371 ℕcn 12168 2c2 12230 ℕ₀cn0 12431 ℤcz 12518 ℤ_≥cuz 12782 ...cfz 13455 APcvdwa 16930 MonoAP cvdwm 16931 PolyAP cvdwp 16932

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-oadd 8403 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-dju 9819 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-n0 12432 df-z 12519 df-uz 12783 df-rp 12937 df-fz 13456 df-hash 14287 df-vdwap 16933 df-vdwmc 16934 df-vdwpc 16935

This theorem is referenced by: vdwlem10 16955

W3C validator