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Theorem vdwlem7 17021

Description: Lemma for vdw 17028. (Contributed by Mario Carneiro, 12-Sep-2014.)

**Hypotheses**
Ref	Expression
vdwlem3.v	⊢ (𝜑 → 𝑉 ∈ ℕ)
vdwlem3.w	⊢ (𝜑 → 𝑊 ∈ ℕ)
vdwlem4.r	⊢ (𝜑 → 𝑅 ∈ Fin)
vdwlem4.h	⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
vdwlem4.f	⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
vdwlem7.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
vdwlem7.g	⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅)
vdwlem7.k	⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
vdwlem7.a	⊢ (𝜑 → 𝐴 ∈ ℕ)
vdwlem7.d	⊢ (𝜑 → 𝐷 ∈ ℕ)
vdwlem7.s	⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (^◡𝐹 “ {𝐺}))

**Assertion**
Ref	Expression
vdwlem7	⊢ (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐺 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐺,𝑦 𝑥,𝐾,𝑦 𝜑,𝑥,𝑦 𝑥,𝑅,𝑦 𝑥,𝐻,𝑦 𝑥,𝑀,𝑦 𝑥,𝐷,𝑦 𝑥,𝑊,𝑦 𝑥,𝑉,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦)

Proof of Theorem vdwlem7 Dummy variables 𝑘 𝑎 𝑑 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7464	. . 3 ⊢ (1...𝑊) ∈ V
2		2nn0 12541	. . . 4 ⊢ 2 ∈ ℕ₀
3		vdwlem7.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
4		eluznn0 12957	. . . 4 ⊢ ((2 ∈ ℕ₀ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℕ₀)
5	2, 3, 4	sylancr 587	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
6		vdwlem7.g	. . 3 ⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅)
7		vdwlem7.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8		eqid 2735	. . 3 ⊢ (1...𝑀) = (1...𝑀)
9	1, 5, 6, 7, 8	vdwpc 17014	. 2 ⊢ (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐺 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_m (1...𝑀))(∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)))
10		vdwlem3.v	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ ℕ)
11	10	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑉 ∈ ℕ)
12		vdwlem3.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℕ)
13	12	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑊 ∈ ℕ)
14		vdwlem4.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Fin)
15	14	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑅 ∈ Fin)
16		vdwlem4.h	. . . . . 6 ⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
17	16	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
18		vdwlem4.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
19	7	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑀 ∈ ℕ)
20	6	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝐺:(1...𝑊)⟶𝑅)
21	3	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝐾 ∈ (ℤ_≥‘2))
22		vdwlem7.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℕ)
23	22	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝐴 ∈ ℕ)
24		vdwlem7.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℕ)
25	24	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝐷 ∈ ℕ)
26		vdwlem7.s	. . . . . 6 ⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (^◡𝐹 “ {𝐺}))
27	26	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → (𝐴(AP‘𝐾)𝐷) ⊆ (^◡𝐹 “ {𝐺}))
28		simplrl 777	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑎 ∈ ℕ)
29		simplrr 778	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑑 ∈ (ℕ ↑_m (1...𝑀)))
30		nnex 12270	. . . . . . 7 ⊢ ℕ ∈ V
31		ovex 7464	. . . . . . 7 ⊢ (1...𝑀) ∈ V
32	30, 31	elmap 8910	. . . . . 6 ⊢ (𝑑 ∈ (ℕ ↑_m (1...𝑀)) ↔ 𝑑:(1...𝑀)⟶ℕ)
33	29, 32	sylib 218	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑑:(1...𝑀)⟶ℕ)
34		simprl 771	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → ∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}))
35		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑑‘𝑖) = (𝑑‘𝑘))
36	35	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑎 + (𝑑‘𝑖)) = (𝑎 + (𝑑‘𝑘)))
37	36, 35	oveq12d 7449	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) = ((𝑎 + (𝑑‘𝑘))(AP‘𝐾)(𝑑‘𝑘)))
38	36	fveq2d 6911	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝐺‘(𝑎 + (𝑑‘𝑖))) = (𝐺‘(𝑎 + (𝑑‘𝑘))))
39	38	sneqd 4643	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → {(𝐺‘(𝑎 + (𝑑‘𝑖)))} = {(𝐺‘(𝑎 + (𝑑‘𝑘)))})
40	39	imaeq2d 6080	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) = (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑘)))}))
41	37, 40	sseq12d 4029	. . . . . . 7 ⊢ (𝑖 = 𝑘 → (((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ↔ ((𝑎 + (𝑑‘𝑘))(AP‘𝐾)(𝑑‘𝑘)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑘)))})))
42	41	cbvralvw 3235	. . . . . 6 ⊢ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ↔ ∀𝑘 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑘))(AP‘𝐾)(𝑑‘𝑘)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑘)))}))
43	34, 42	sylib 218	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → ∀𝑘 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑘))(AP‘𝐾)(𝑑‘𝑘)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑘)))}))
44	38	cbvmptv 5261	. . . . 5 ⊢ (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖)))) = (𝑘 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑘))))
45		simprr 773	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)
46		eqid 2735	. . . . 5 ⊢ (𝑎 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) = (𝑎 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)))
47		eqid 2735	. . . . 5 ⊢ (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝑑‘𝑗)) + (𝑊 · 𝐷))) = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝑑‘𝑗)) + (𝑊 · 𝐷)))
48	11, 13, 15, 17, 18, 19, 20, 21, 23, 25, 27, 28, 33, 43, 44, 45, 46, 47	vdwlem6 17020	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
49	48	ex 412	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...𝑀)))) → ((∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))
50	49	rexlimdvva 3211	. 2 ⊢ (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_m (1...𝑀))(∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))
51	9, 50	sylbid 240	1 ⊢ (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐺 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 ifcif 4531 {csn 4631 ⟨cop 4637 class class class wbr 5148 ↦ cmpt 5231 ^◡ccnv 5688 ran crn 5690 “ cima 5692 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 Fincfn 8984 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 ℕcn 12264 2c2 12319 ℕ₀cn0 12524 ℤ_≥cuz 12876 ...cfz 13544 ♯chash 14366 APcvdwa 16999 MonoAP cvdwm 17000 PolyAP cvdwp 17001

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-hash 14367 df-vdwap 17002 df-vdwmc 17003 df-vdwpc 17004

This theorem is referenced by: vdwlem9 17023

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