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Theorem vdwlem7 16851
Description: Lemma for vdw 16858. (Contributed by Mario Carneiro, 12-Sep-2014.)
Hypotheses
Ref Expression
vdwlem3.v (𝜑𝑉 ∈ ℕ)
vdwlem3.w (𝜑𝑊 ∈ ℕ)
vdwlem4.r (𝜑𝑅 ∈ Fin)
vdwlem4.h (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
vdwlem4.f 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
vdwlem7.m (𝜑𝑀 ∈ ℕ)
vdwlem7.g (𝜑𝐺:(1...𝑊)⟶𝑅)
vdwlem7.k (𝜑𝐾 ∈ (ℤ‘2))
vdwlem7.a (𝜑𝐴 ∈ ℕ)
vdwlem7.d (𝜑𝐷 ∈ 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.
StepHypRef Expression
1 ovex 7386 . . 3 (1...𝑊) ∈ V
2 2nn0 12426 . . . 4 2 ∈ ℕ0
3 vdwlem7.k . . . 4 (𝜑𝐾 ∈ (ℤ‘2))
4 eluznn0 12834 . . . 4 ((2 ∈ ℕ0𝐾 ∈ (ℤ‘2)) → 𝐾 ∈ ℕ0)
52, 3, 4sylancr 587 . . 3 (𝜑𝐾 ∈ ℕ0)
6 vdwlem7.g . . 3 (𝜑𝐺:(1...𝑊)⟶𝑅)
7 vdwlem7.m . . 3 (𝜑𝑀 ∈ ℕ)
8 eqid 2736 . . 3 (1...𝑀) = (1...𝑀)
91, 5, 6, 7, 8vdwpc 16844 . 2 (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐺 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m (1...𝑀))(∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)))
10 vdwlem3.v . . . . . 6 (𝜑𝑉 ∈ ℕ)
1110ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → 𝑉 ∈ ℕ)
12 vdwlem3.w . . . . . 6 (𝜑𝑊 ∈ ℕ)
1312ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → 𝑊 ∈ ℕ)
14 vdwlem4.r . . . . . 6 (𝜑𝑅 ∈ Fin)
1514ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → 𝑅 ∈ Fin)
16 vdwlem4.h . . . . . 6 (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
1716ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
18 vdwlem4.f . . . . 5 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
197ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → 𝑀 ∈ ℕ)
206ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → 𝐺:(1...𝑊)⟶𝑅)
213ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → 𝐾 ∈ (ℤ‘2))
22 vdwlem7.a . . . . . 6 (𝜑𝐴 ∈ ℕ)
2322ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → 𝐴 ∈ ℕ)
24 vdwlem7.d . . . . . 6 (𝜑𝐷 ∈ ℕ)
2524ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → 𝐷 ∈ ℕ)
26 vdwlem7.s . . . . . 6 (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐹 “ {𝐺}))
2726ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → (𝐴(AP‘𝐾)𝐷) ⊆ (𝐹 “ {𝐺}))
28 simplrl 775 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → 𝑎 ∈ ℕ)
29 simplrr 776 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → 𝑑 ∈ (ℕ ↑m (1...𝑀)))
30 nnex 12155 . . . . . . 7 ℕ ∈ V
31 ovex 7386 . . . . . . 7 (1...𝑀) ∈ V
3230, 31elmap 8805 . . . . . 6 (𝑑 ∈ (ℕ ↑m (1...𝑀)) ↔ 𝑑:(1...𝑀)⟶ℕ)
3329, 32sylib 217 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → 𝑑:(1...𝑀)⟶ℕ)
34 simprl 769 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → ∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}))
35 fveq2 6839 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝑑𝑖) = (𝑑𝑘))
3635oveq2d 7369 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑎 + (𝑑𝑖)) = (𝑎 + (𝑑𝑘)))
3736, 35oveq12d 7371 . . . . . . . 8 (𝑖 = 𝑘 → ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) = ((𝑎 + (𝑑𝑘))(AP‘𝐾)(𝑑𝑘)))
3836fveq2d 6843 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝐺‘(𝑎 + (𝑑𝑖))) = (𝐺‘(𝑎 + (𝑑𝑘))))
3938sneqd 4596 . . . . . . . . 9 (𝑖 = 𝑘 → {(𝐺‘(𝑎 + (𝑑𝑖)))} = {(𝐺‘(𝑎 + (𝑑𝑘)))})
4039imaeq2d 6011 . . . . . . . 8 (𝑖 = 𝑘 → (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) = (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑘)))}))
4137, 40sseq12d 3975 . . . . . . 7 (𝑖 = 𝑘 → (((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ↔ ((𝑎 + (𝑑𝑘))(AP‘𝐾)(𝑑𝑘)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑘)))})))
4241cbvralvw 3223 . . . . . 6 (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ↔ ∀𝑘 ∈ (1...𝑀)((𝑎 + (𝑑𝑘))(AP‘𝐾)(𝑑𝑘)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑘)))}))
4334, 42sylib 217 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → ∀𝑘 ∈ (1...𝑀)((𝑎 + (𝑑𝑘))(AP‘𝐾)(𝑑𝑘)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑘)))}))
4438cbvmptv 5216 . . . . 5 (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖)))) = (𝑘 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑘))))
45 simprr 771 . . . . 5 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)
46 eqid 2736 . . . . 5 (𝑎 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) = (𝑎 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)))
47 eqid 2736 . . . . 5 (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝑑𝑗)) + (𝑊 · 𝐷))) = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝑑𝑗)) + (𝑊 · 𝐷)))
4811, 13, 15, 17, 18, 19, 20, 21, 23, 25, 27, 28, 33, 43, 44, 45, 46, 47vdwlem6 16850 . . . 4 (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀)) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
4948ex 413 . . 3 ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑m (1...𝑀)))) → ((∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))
5049rexlimdvva 3203 . 2 (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑m (1...𝑀))(∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ∧ (♯‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖))))) = 𝑀) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))
519, 50sylbid 239 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 845   = wceq 1541  wcel 2106  wral 3062  wrex 3071  wss 3908  ifcif 4484  {csn 4584  cop 4590   class class class wbr 5103  cmpt 5186  ccnv 5630  ran crn 5632  cima 5634  wf 6489  cfv 6493  (class class class)co 7353  m cmap 8761  Fincfn 8879  0cc0 11047  1c1 11048   + caddc 11050   · cmul 11052  cmin 11381  cn 12149  2c2 12204  0cn0 12409  cuz 12759  ...cfz 13416  chash 14222  APcvdwa 16829   MonoAP cvdwm 16830   PolyAP cvdwp 16831
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-1o 8408  df-oadd 8412  df-er 8644  df-map 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-dju 9833  df-card 9871  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-nn 12150  df-2 12212  df-n0 12410  df-z 12496  df-uz 12760  df-rp 12908  df-fz 13417  df-hash 14223  df-vdwap 16832  df-vdwmc 16833  df-vdwpc 16834
This theorem is referenced by:  vdwlem9  16853
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