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Theorem vdwlem7 17025
Description: Lemma for vdw 17032. (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 7464 . . 3 (1...𝑊) ∈ V
2 2nn0 12543 . . . 4 2 ∈ ℕ0
3 vdwlem7.k . . . 4 (𝜑𝐾 ∈ (ℤ‘2))
4 eluznn0 12959 . . . 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 2737 . . 3 (1...𝑀) = (1...𝑀)
91, 5, 6, 7, 8vdwpc 17018 . 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 726 . . . . 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 726 . . . . 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 726 . . . . 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 726 . . . . 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 726 . . . . 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 726 . . . . 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 726 . . . . 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 726 . . . . 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 726 . . . . 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 726 . . . . 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 777 . . . . 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 778 . . . . . 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 12272 . . . . . . 7 ℕ ∈ V
31 ovex 7464 . . . . . . 7 (1...𝑀) ∈ V
3230, 31elmap 8911 . . . . . 6 (𝑑 ∈ (ℕ ↑m (1...𝑀)) ↔ 𝑑:(1...𝑀)⟶ℕ)
3329, 32sylib 218 . . . . 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 771 . . . . . 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 6906 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝑑𝑖) = (𝑑𝑘))
3635oveq2d 7447 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑎 + (𝑑𝑖)) = (𝑎 + (𝑑𝑘)))
3736, 35oveq12d 7449 . . . . . . . 8 (𝑖 = 𝑘 → ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) = ((𝑎 + (𝑑𝑘))(AP‘𝐾)(𝑑𝑘)))
3836fveq2d 6910 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝐺‘(𝑎 + (𝑑𝑖))) = (𝐺‘(𝑎 + (𝑑𝑘))))
3938sneqd 4638 . . . . . . . . 9 (𝑖 = 𝑘 → {(𝐺‘(𝑎 + (𝑑𝑖)))} = {(𝐺‘(𝑎 + (𝑑𝑘)))})
4039imaeq2d 6078 . . . . . . . 8 (𝑖 = 𝑘 → (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) = (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑘)))}))
4137, 40sseq12d 4017 . . . . . . 7 (𝑖 = 𝑘 → (((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ↔ ((𝑎 + (𝑑𝑘))(AP‘𝐾)(𝑑𝑘)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑘)))})))
4241cbvralvw 3237 . . . . . 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 218 . . . . 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 5255 . . . . 5 (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖)))) = (𝑘 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑘))))
45 simprr 773 . . . . 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 2737 . . . . 5 (𝑎 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) = (𝑎 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)))
47 eqid 2737 . . . . 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 17024 . . . 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 412 . . 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 3213 . 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 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 848   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951  ifcif 4525  {csn 4626  cop 4632   class class class wbr 5143  cmpt 5225  ccnv 5684  ran crn 5686  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  Fincfn 8985  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  cmin 11492  cn 12266  2c2 12321  0cn0 12526  cuz 12878  ...cfz 13547  chash 14369  APcvdwa 17003   MonoAP cvdwm 17004   PolyAP cvdwp 17005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-hash 14370  df-vdwap 17006  df-vdwmc 17007  df-vdwpc 17008
This theorem is referenced by:  vdwlem9  17027
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