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Theorem vdwlem7 17034

Description: Lemma for vdw 17041. (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 7481	. . 3 ⊢ (1...𝑊) ∈ V
2		2nn0 12570	. . . 4 ⊢ 2 ∈ ℕ₀
3		vdwlem7.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
4		eluznn0 12982	. . . 4 ⊢ ((2 ∈ ℕ₀ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℕ₀)
5	2, 3, 4	sylancr 586	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
6		vdwlem7.g	. . 3 ⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅)
7		vdwlem7.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8		eqid 2740	. . 3 ⊢ (1...𝑀) = (1...𝑀)
9	1, 5, 6, 7, 8	vdwpc 17027	. 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 725	. . . . 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 725	. . . . 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 725	. . . . 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 725	. . . . 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 725	. . . . 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 725	. . . . 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 725	. . . . 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 725	. . . . 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 725	. . . . 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 725	. . . . 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 776	. . . . 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 777	. . . . . 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 12299	. . . . . . 7 ⊢ ℕ ∈ V
31		ovex 7481	. . . . . . 7 ⊢ (1...𝑀) ∈ V
32	30, 31	elmap 8929	. . . . . 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 770	. . . . . 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 6920	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑑‘𝑖) = (𝑑‘𝑘))
36	35	oveq2d 7464	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑎 + (𝑑‘𝑖)) = (𝑎 + (𝑑‘𝑘)))
37	36, 35	oveq12d 7466	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) = ((𝑎 + (𝑑‘𝑘))(AP‘𝐾)(𝑑‘𝑘)))
38	36	fveq2d 6924	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝐺‘(𝑎 + (𝑑‘𝑖))) = (𝐺‘(𝑎 + (𝑑‘𝑘))))
39	38	sneqd 4660	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → {(𝐺‘(𝑎 + (𝑑‘𝑖)))} = {(𝐺‘(𝑎 + (𝑑‘𝑘)))})
40	39	imaeq2d 6089	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) = (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑘)))}))
41	37, 40	sseq12d 4042	. . . . . . 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 3243	. . . . . 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 5279	. . . . 5 ⊢ (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖)))) = (𝑘 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑘))))
45		simprr 772	. . . . 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 2740	. . . . 5 ⊢ (𝑎 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) = (𝑎 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)))
47		eqid 2740	. . . . 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 17033	. . . 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 3219	. 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 846 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 ⊆ wss 3976 ifcif 4548 {csn 4648 ⟨cop 4654 class class class wbr 5166 ↦ cmpt 5249 ^◡ccnv 5699 ran crn 5701 “ cima 5703 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 Fincfn 9003 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 − cmin 11520 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤ_≥cuz 12903 ...cfz 13567 ♯chash 14379 APcvdwa 17012 MonoAP cvdwm 17013 PolyAP cvdwp 17014

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-oadd 8526 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fz 13568 df-hash 14380 df-vdwap 17015 df-vdwmc 17016 df-vdwpc 17017

This theorem is referenced by: vdwlem9 17036

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