MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vdwlem7 Structured version   Visualization version   GIF version

Theorem vdwlem7 16311
Description: Lemma for vdw 16318. (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 7178 . . 3 (1...𝑊) ∈ V
2 2nn0 11902 . . . 4 2 ∈ ℕ0
3 vdwlem7.k . . . 4 (𝜑𝐾 ∈ (ℤ‘2))
4 eluznn0 12305 . . . 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 2818 . . 3 (1...𝑀) = (1...𝑀)
91, 5, 6, 7, 8vdwpc 16304 . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 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 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)) → 𝑎 ∈ ℕ)
29 simplrr 774 . . . . . 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 11632 . . . . . . 7 ℕ ∈ V
31 ovex 7178 . . . . . . 7 (1...𝑀) ∈ V
3230, 31elmap 8424 . . . . . 6 (𝑑 ∈ (ℕ ↑m (1...𝑀)) ↔ 𝑑:(1...𝑀)⟶ℕ)
3329, 32sylib 219 . . . . 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 767 . . . . . 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 6663 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝑑𝑖) = (𝑑𝑘))
3635oveq2d 7161 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑎 + (𝑑𝑖)) = (𝑎 + (𝑑𝑘)))
3736, 35oveq12d 7163 . . . . . . . 8 (𝑖 = 𝑘 → ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) = ((𝑎 + (𝑑𝑘))(AP‘𝐾)(𝑑𝑘)))
3836fveq2d 6667 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝐺‘(𝑎 + (𝑑𝑖))) = (𝐺‘(𝑎 + (𝑑𝑘))))
3938sneqd 4569 . . . . . . . . 9 (𝑖 = 𝑘 → {(𝐺‘(𝑎 + (𝑑𝑖)))} = {(𝐺‘(𝑎 + (𝑑𝑘)))})
4039imaeq2d 5922 . . . . . . . 8 (𝑖 = 𝑘 → (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) = (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑘)))}))
4137, 40sseq12d 3997 . . . . . . 7 (𝑖 = 𝑘 → (((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ↔ ((𝑎 + (𝑑𝑘))(AP‘𝐾)(𝑑𝑘)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑘)))})))
4241cbvralvw 3447 . . . . . 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 219 . . . . 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 5160 . . . . 5 (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖)))) = (𝑘 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑘))))
45 simprr 769 . . . . 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 2818 . . . . 5 (𝑎 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) = (𝑎 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)))
47 eqid 2818 . . . . 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 16310 . . . 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 3291 . 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 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 841   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933  ifcif 4463  {csn 4557  cop 4563   class class class wbr 5057  cmpt 5137  ccnv 5547  ran crn 5549  cima 5551  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  Fincfn 8497  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530  cmin 10858  cn 11626  2c2 11680  0cn0 11885  cuz 12231  ...cfz 12880  chash 13678  APcvdwa 16289   MonoAP cvdwm 16290   PolyAP cvdwp 16291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-hash 13679  df-vdwap 16292  df-vdwmc 16293  df-vdwpc 16294
This theorem is referenced by:  vdwlem9  16313
  Copyright terms: Public domain W3C validator