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Theorem vdwlem7 15898
Description: Lemma for vdw 15905. (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 6827 . . 3 (1...𝑊) ∈ V
2 2nn0 11516 . . . 4 2 ∈ ℕ0
3 vdwlem7.k . . . 4 (𝜑𝐾 ∈ (ℤ‘2))
4 eluznn0 11965 . . . 4 ((2 ∈ ℕ0𝐾 ∈ (ℤ‘2)) → 𝐾 ∈ ℕ0)
52, 3, 4sylancr 575 . . 3 (𝜑𝐾 ∈ ℕ0)
6 vdwlem7.g . . 3 (𝜑𝐺:(1...𝑊)⟶𝑅)
7 vdwlem7.m . . 3 (𝜑𝑀 ∈ ℕ)
8 eqid 2771 . . 3 (1...𝑀) = (1...𝑀)
91, 5, 6, 7, 8vdwpc 15891 . 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 705 . . . . 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 705 . . . . 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 705 . . . . 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 705 . . . . 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 705 . . . . 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 705 . . . . 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 705 . . . . 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 705 . . . . 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 705 . . . . 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 705 . . . . 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 762 . . . . 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 763 . . . . . 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 11232 . . . . . . 7 ℕ ∈ V
31 ovex 6827 . . . . . . 7 (1...𝑀) ∈ V
3230, 31elmap 8042 . . . . . 6 (𝑑 ∈ (ℕ ↑𝑚 (1...𝑀)) ↔ 𝑑:(1...𝑀)⟶ℕ)
3329, 32sylib 208 . . . . 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 754 . . . . . 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 6333 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝑑𝑖) = (𝑑𝑘))
3635oveq2d 6812 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑎 + (𝑑𝑖)) = (𝑎 + (𝑑𝑘)))
3736, 35oveq12d 6814 . . . . . . . 8 (𝑖 = 𝑘 → ((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) = ((𝑎 + (𝑑𝑘))(AP‘𝐾)(𝑑𝑘)))
3836fveq2d 6337 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝐺‘(𝑎 + (𝑑𝑖))) = (𝐺‘(𝑎 + (𝑑𝑘))))
3938sneqd 4329 . . . . . . . . 9 (𝑖 = 𝑘 → {(𝐺‘(𝑎 + (𝑑𝑖)))} = {(𝐺‘(𝑎 + (𝑑𝑘)))})
4039imaeq2d 5606 . . . . . . . 8 (𝑖 = 𝑘 → (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) = (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑘)))}))
4137, 40sseq12d 3783 . . . . . . 7 (𝑖 = 𝑘 → (((𝑎 + (𝑑𝑖))(AP‘𝐾)(𝑑𝑖)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑖)))}) ↔ ((𝑎 + (𝑑𝑘))(AP‘𝐾)(𝑑𝑘)) ⊆ (𝐺 “ {(𝐺‘(𝑎 + (𝑑𝑘)))})))
4241cbvralv 3320 . . . . . 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 208 . . . . 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 4885 . . . . 5 (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑖)))) = (𝑘 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑𝑘))))
45 simprr 756 . . . . 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 2771 . . . . 5 (𝑎 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1))) = (𝑎 + (𝑊 · ((𝐴 + (𝑉𝐷)) − 1)))
47 eqid 2771 . . . . 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 15897 . . . 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 397 . . 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 3186 . 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 230 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 382  wo 836   = wceq 1631  wcel 2145  wral 3061  wrex 3062  wss 3723  ifcif 4226  {csn 4317  cop 4323   class class class wbr 4787  cmpt 4864  ccnv 5249  ran crn 5251  cima 5253  wf 6026  cfv 6030  (class class class)co 6796  𝑚 cmap 8013  Fincfn 8113  0cc0 10142  1c1 10143   + caddc 10145   · cmul 10147  cmin 10472  cn 11226  2c2 11276  0cn0 11499  cuz 11893  ...cfz 12533  chash 13321  APcvdwa 15876   MonoAP cvdwm 15877   PolyAP cvdwp 15878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-map 8015  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8969  df-cda 9196  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-n0 11500  df-z 11585  df-uz 11894  df-rp 12036  df-fz 12534  df-hash 13322  df-vdwap 15879  df-vdwmc 15880  df-vdwpc 15881
This theorem is referenced by:  vdwlem9  15900
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