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Theorem vdwlem7 16949

Description: Lemma for vdw 16956. (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 7389	. . 3 ⊢ (1...𝑊) ∈ V
2		2nn0 12445	. . . 4 ⊢ 2 ∈ ℕ₀
3		vdwlem7.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
4		eluznn0 12858	. . . 4 ⊢ ((2 ∈ ℕ₀ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℕ₀)
5	2, 3, 4	sylancr 593	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
6		vdwlem7.g	. . 3 ⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅)
7		vdwlem7.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8		eqid 2739	. . 3 ⊢ (1...𝑀) = (1...𝑀)
9	1, 5, 6, 7, 8	vdwpc 16942	. 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 732	. . . . 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 732	. . . . 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 732	. . . . 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 732	. . . . 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 732	. . . . 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 732	. . . . 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 732	. . . . 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 732	. . . . 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 732	. . . . 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 732	. . . . 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 782	. . . . 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 783	. . . . . 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 12171	. . . . . . 7 ⊢ ℕ ∈ V
31		ovex 7389	. . . . . . 7 ⊢ (1...𝑀) ∈ V
32	30, 31	elmap 8809	. . . . . 6 ⊢ (𝑑 ∈ (ℕ ↑_m (1...𝑀)) ↔ 𝑑:(1...𝑀)⟶ℕ)
33	29, 32	sylib 219	. . . . 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 776	. . . . . 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 6827	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑑‘𝑖) = (𝑑‘𝑘))
36	35	oveq2d 7372	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑎 + (𝑑‘𝑖)) = (𝑎 + (𝑑‘𝑘)))
37	36, 35	oveq12d 7374	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) = ((𝑎 + (𝑑‘𝑘))(AP‘𝐾)(𝑑‘𝑘)))
38	36	fveq2d 6831	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝐺‘(𝑎 + (𝑑‘𝑖))) = (𝐺‘(𝑎 + (𝑑‘𝑘))))
39	38	sneqd 4567	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → {(𝐺‘(𝑎 + (𝑑‘𝑖)))} = {(𝐺‘(𝑎 + (𝑑‘𝑘)))})
40	39	imaeq2d 6012	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) = (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑘)))}))
41	37, 40	sseq12d 3948	. . . . . . 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 3217	. . . . . 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 219	. . . . 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 5176	. . . . 5 ⊢ (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖)))) = (𝑘 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑘))))
45		simprr 778	. . . . 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 2739	. . . . 5 ⊢ (𝑎 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) = (𝑎 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)))
47		eqid 2739	. . . . 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 16948	. . . 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 413	. . 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 3196	. 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 241	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 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∃wrex 3063 ⊆ wss 3883 ifcif 4454 {csn 4555 ⟨cop 4561 class class class wbr 5072 ↦ cmpt 5153 ^◡ccnv 5617 ran crn 5619 “ cima 5621 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ↑_m cmap 8763 Fincfn 8883 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 − cmin 11368 ℕcn 12165 2c2 12227 ℕ₀cn0 12428 ℤ_≥cuz 12779 ...cfz 13452 ♯chash 14283 APcvdwa 16927 MonoAP cvdwm 16928 PolyAP cvdwp 16929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-fz 13453 df-hash 14284 df-vdwap 16930 df-vdwmc 16931 df-vdwpc 16932

This theorem is referenced by: vdwlem9 16951

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