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Theorem vdwlem9 17023
Description: Lemma for vdw 17028. (Contributed by Mario Carneiro, 12-Sep-2014.)
Hypotheses
Ref Expression
vdw.r (𝜑𝑅 ∈ Fin)
vdwlem9.k (𝜑𝐾 ∈ (ℤ‘2))
vdwlem9.s (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝐾 MonoAP 𝑓)
vdwlem9.m (𝜑𝑀 ∈ ℕ)
vdwlem9.w (𝜑𝑊 ∈ ℕ)
vdwlem9.g (𝜑 → ∀𝑔 ∈ (𝑅m (1...𝑊))(⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
vdwlem9.v (𝜑𝑉 ∈ ℕ)
vdwlem9.a (𝜑 → ∀𝑓 ∈ ((𝑅m (1...𝑊)) ↑m (1...𝑉))𝐾 MonoAP 𝑓)
vdwlem9.h (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
vdwlem9.f 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
Assertion
Ref Expression
vdwlem9 (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻))
Distinct variable groups:   𝑔,𝑛,𝑥,𝑦,𝜑   𝑥,𝑓,𝑦,𝑉   𝑓,𝑊,𝑥,𝑦   𝑓,𝑔,𝐹,𝑥,𝑦   𝑓,𝑛,𝑠,𝐾,𝑔,𝑥,𝑦   𝑓,𝑀,𝑔,𝑛,𝑥,𝑦   𝑅,𝑓,𝑔,𝑛,𝑠,𝑥,𝑦   𝑔,𝐻,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑓,𝑠)   𝐹(𝑛,𝑠)   𝐻(𝑓,𝑛,𝑠)   𝑀(𝑠)   𝑉(𝑔,𝑛,𝑠)   𝑊(𝑔,𝑛,𝑠)

Proof of Theorem vdwlem9
Dummy variables 𝑎 𝑑 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 5152 . . 3 (𝑓 = 𝐹 → (𝐾 MonoAP 𝑓𝐾 MonoAP 𝐹))
2 vdwlem9.a . . 3 (𝜑 → ∀𝑓 ∈ ((𝑅m (1...𝑊)) ↑m (1...𝑉))𝐾 MonoAP 𝑓)
3 vdwlem9.v . . . . 5 (𝜑𝑉 ∈ ℕ)
4 vdwlem9.w . . . . 5 (𝜑𝑊 ∈ ℕ)
5 vdw.r . . . . 5 (𝜑𝑅 ∈ Fin)
6 vdwlem9.h . . . . 5 (𝜑𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
7 vdwlem9.f . . . . 5 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
83, 4, 5, 6, 7vdwlem4 17018 . . . 4 (𝜑𝐹:(1...𝑉)⟶(𝑅m (1...𝑊)))
9 ovex 7464 . . . . 5 (𝑅m (1...𝑊)) ∈ V
10 ovex 7464 . . . . 5 (1...𝑉) ∈ V
119, 10elmap 8910 . . . 4 (𝐹 ∈ ((𝑅m (1...𝑊)) ↑m (1...𝑉)) ↔ 𝐹:(1...𝑉)⟶(𝑅m (1...𝑊)))
128, 11sylibr 234 . . 3 (𝜑𝐹 ∈ ((𝑅m (1...𝑊)) ↑m (1...𝑉)))
131, 2, 12rspcdva 3623 . 2 (𝜑𝐾 MonoAP 𝐹)
14 vdwlem9.k . . . . . 6 (𝜑𝐾 ∈ (ℤ‘2))
15 eluz2nn 12922 . . . . . 6 (𝐾 ∈ (ℤ‘2) → 𝐾 ∈ ℕ)
1614, 15syl 17 . . . . 5 (𝜑𝐾 ∈ ℕ)
1716nnnn0d 12585 . . . 4 (𝜑𝐾 ∈ ℕ0)
1810, 17, 8vdwmc 17012 . . 3 (𝜑 → (𝐾 MonoAP 𝐹 ↔ ∃𝑔𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔})))
19 vdwlem9.g . . . . . . . . 9 (𝜑 → ∀𝑔 ∈ (𝑅m (1...𝑊))(⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
2019adantr 480 . . . . . . . 8 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ∀𝑔 ∈ (𝑅m (1...𝑊))(⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
21 simprr 773 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))
2216adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐾 ∈ ℕ)
23 simprll 779 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ ℕ)
24 simprlr 780 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑑 ∈ ℕ)
25 vdwapid1 17009 . . . . . . . . . . . . 13 ((𝐾 ∈ ℕ ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → 𝑎 ∈ (𝑎(AP‘𝐾)𝑑))
2622, 23, 24, 25syl3anc 1370 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ (𝑎(AP‘𝐾)𝑑))
2721, 26sseldd 3996 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ (𝐹 “ {𝑔}))
288ffnd 6738 . . . . . . . . . . . . 13 (𝜑𝐹 Fn (1...𝑉))
2928adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐹 Fn (1...𝑉))
30 fniniseg 7080 . . . . . . . . . . . 12 (𝐹 Fn (1...𝑉) → (𝑎 ∈ (𝐹 “ {𝑔}) ↔ (𝑎 ∈ (1...𝑉) ∧ (𝐹𝑎) = 𝑔)))
3129, 30syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 ∈ (𝐹 “ {𝑔}) ↔ (𝑎 ∈ (1...𝑉) ∧ (𝐹𝑎) = 𝑔)))
3227, 31mpbid 232 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 ∈ (1...𝑉) ∧ (𝐹𝑎) = 𝑔))
3332simprd 495 . . . . . . . . 9 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝐹𝑎) = 𝑔)
348adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐹:(1...𝑉)⟶(𝑅m (1...𝑊)))
3532simpld 494 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ (1...𝑉))
3634, 35ffvelcdmd 7105 . . . . . . . . 9 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝐹𝑎) ∈ (𝑅m (1...𝑊)))
3733, 36eqeltrrd 2840 . . . . . . . 8 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑔 ∈ (𝑅m (1...𝑊)))
38 rsp 3245 . . . . . . . 8 (∀𝑔 ∈ (𝑅m (1...𝑊))(⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) → (𝑔 ∈ (𝑅m (1...𝑊)) → (⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔)))
3920, 37, 38sylc 65 . . . . . . 7 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔))
403adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑉 ∈ ℕ)
414adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑊 ∈ ℕ)
425adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑅 ∈ Fin)
436adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
44 vdwlem9.m . . . . . . . . . . 11 (𝜑𝑀 ∈ ℕ)
4544adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑀 ∈ ℕ)
46 ovex 7464 . . . . . . . . . . . 12 (1...𝑊) ∈ V
47 elmapg 8878 . . . . . . . . . . . 12 ((𝑅 ∈ Fin ∧ (1...𝑊) ∈ V) → (𝑔 ∈ (𝑅m (1...𝑊)) ↔ 𝑔:(1...𝑊)⟶𝑅))
4842, 46, 47sylancl 586 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑔 ∈ (𝑅m (1...𝑊)) ↔ 𝑔:(1...𝑊)⟶𝑅))
4937, 48mpbid 232 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑔:(1...𝑊)⟶𝑅)
5014adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐾 ∈ (ℤ‘2))
5140, 41, 42, 43, 7, 45, 49, 50, 23, 24, 21vdwlem7 17021 . . . . . . . . 9 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (⟨𝑀, 𝐾⟩ PolyAP 𝑔 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝑔)))
52 olc 868 . . . . . . . . . 10 ((𝐾 + 1) MonoAP 𝑔 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝑔))
5352a1i 11 . . . . . . . . 9 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝐾 + 1) MonoAP 𝑔 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝑔)))
5451, 53jaod 859 . . . . . . . 8 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝑔)))
55 oveq1 7438 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥 − 1) = (𝑎 − 1))
5655oveq1d 7446 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥 − 1) + 𝑉) = ((𝑎 − 1) + 𝑉))
5756oveq2d 7447 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (𝑊 · ((𝑥 − 1) + 𝑉)) = (𝑊 · ((𝑎 − 1) + 𝑉)))
5857oveq2d 7447 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))) = (𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))
5958fveq2d 6911 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
6059mpteq2dv 5250 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))))
6146mptex 7243 . . . . . . . . . . . . . 14 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) ∈ V
6260, 7, 61fvmpt 7016 . . . . . . . . . . . . 13 (𝑎 ∈ (1...𝑉) → (𝐹𝑎) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))))
6335, 62syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝐹𝑎) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))))
6463, 33eqtr3d 2777 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) = 𝑔)
6564breq2d 5160 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝐾 + 1) MonoAP (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) ↔ (𝐾 + 1) MonoAP 𝑔))
6617adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐾 ∈ ℕ0)
67 peano2nn0 12564 . . . . . . . . . . . 12 (𝐾 ∈ ℕ0 → (𝐾 + 1) ∈ ℕ0)
6866, 67syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝐾 + 1) ∈ ℕ0)
69 nnm1nn0 12565 . . . . . . . . . . . . . 14 (𝑎 ∈ ℕ → (𝑎 − 1) ∈ ℕ0)
7023, 69syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 − 1) ∈ ℕ0)
71 nn0nnaddcl 12555 . . . . . . . . . . . . 13 (((𝑎 − 1) ∈ ℕ0𝑉 ∈ ℕ) → ((𝑎 − 1) + 𝑉) ∈ ℕ)
7270, 40, 71syl2anc 584 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝑎 − 1) + 𝑉) ∈ ℕ)
7341, 72nnmulcld 12317 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · ((𝑎 − 1) + 𝑉)) ∈ ℕ)
7423, 40nnaddcld 12316 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ∈ ℕ)
7541, 74nnmulcld 12317 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ∈ ℕ)
7675nnzd 12638 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ∈ ℤ)
77 2nn 12337 . . . . . . . . . . . . . . . . 17 2 ∈ ℕ
78 nnmulcl 12288 . . . . . . . . . . . . . . . . 17 ((2 ∈ ℕ ∧ 𝑉 ∈ ℕ) → (2 · 𝑉) ∈ ℕ)
7977, 3, 78sylancr 587 . . . . . . . . . . . . . . . 16 (𝜑 → (2 · 𝑉) ∈ ℕ)
804, 79nnmulcld 12317 . . . . . . . . . . . . . . 15 (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℕ)
8180nnzd 12638 . . . . . . . . . . . . . 14 (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℤ)
8281adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ ℤ)
8323nnred 12279 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ ℝ)
8440nnred 12279 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑉 ∈ ℝ)
85 elfzle2 13565 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ (1...𝑉) → 𝑎𝑉)
8635, 85syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎𝑉)
8783, 84, 84, 86leadd1dd 11875 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ≤ (𝑉 + 𝑉))
8840nncnd 12280 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑉 ∈ ℂ)
89882timesd 12507 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (2 · 𝑉) = (𝑉 + 𝑉))
9087, 89breqtrrd 5176 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ≤ (2 · 𝑉))
9174nnred 12279 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ∈ ℝ)
9279nnred 12279 . . . . . . . . . . . . . . . 16 (𝜑 → (2 · 𝑉) ∈ ℝ)
9392adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (2 · 𝑉) ∈ ℝ)
9441nnred 12279 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑊 ∈ ℝ)
9541nngt0d 12313 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 0 < 𝑊)
96 lemul2 12118 . . . . . . . . . . . . . . 15 (((𝑎 + 𝑉) ∈ ℝ ∧ (2 · 𝑉) ∈ ℝ ∧ (𝑊 ∈ ℝ ∧ 0 < 𝑊)) → ((𝑎 + 𝑉) ≤ (2 · 𝑉) ↔ (𝑊 · (𝑎 + 𝑉)) ≤ (𝑊 · (2 · 𝑉))))
9791, 93, 94, 95, 96syl112anc 1373 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝑎 + 𝑉) ≤ (2 · 𝑉) ↔ (𝑊 · (𝑎 + 𝑉)) ≤ (𝑊 · (2 · 𝑉))))
9890, 97mpbid 232 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ≤ (𝑊 · (2 · 𝑉)))
99 eluz2 12882 . . . . . . . . . . . . 13 ((𝑊 · (2 · 𝑉)) ∈ (ℤ‘(𝑊 · (𝑎 + 𝑉))) ↔ ((𝑊 · (𝑎 + 𝑉)) ∈ ℤ ∧ (𝑊 · (2 · 𝑉)) ∈ ℤ ∧ (𝑊 · (𝑎 + 𝑉)) ≤ (𝑊 · (2 · 𝑉))))
10076, 82, 98, 99syl3anbrc 1342 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ (ℤ‘(𝑊 · (𝑎 + 𝑉))))
10141nncnd 12280 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑊 ∈ ℂ)
102 1cnd 11254 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 1 ∈ ℂ)
10370nn0cnd 12587 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 − 1) ∈ ℂ)
104103, 88addcld 11278 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝑎 − 1) + 𝑉) ∈ ℂ)
105101, 102, 104adddid 11283 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (1 + ((𝑎 − 1) + 𝑉))) = ((𝑊 · 1) + (𝑊 · ((𝑎 − 1) + 𝑉))))
106102, 103, 88addassd 11281 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((1 + (𝑎 − 1)) + 𝑉) = (1 + ((𝑎 − 1) + 𝑉)))
107 ax-1cn 11211 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
10823nncnd 12280 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ ℂ)
109 pncan3 11514 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (1 + (𝑎 − 1)) = 𝑎)
110107, 108, 109sylancr 587 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (1 + (𝑎 − 1)) = 𝑎)
111110oveq1d 7446 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((1 + (𝑎 − 1)) + 𝑉) = (𝑎 + 𝑉))
112106, 111eqtr3d 2777 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (1 + ((𝑎 − 1) + 𝑉)) = (𝑎 + 𝑉))
113112oveq2d 7447 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (1 + ((𝑎 − 1) + 𝑉))) = (𝑊 · (𝑎 + 𝑉)))
114101mulridd 11276 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · 1) = 𝑊)
115114oveq1d 7446 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝑊 · 1) + (𝑊 · ((𝑎 − 1) + 𝑉))) = (𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉))))
116105, 113, 1153eqtr3d 2783 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) = (𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉))))
117116fveq2d 6911 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (ℤ‘(𝑊 · (𝑎 + 𝑉))) = (ℤ‘(𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
118100, 117eleqtrd 2841 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ (ℤ‘(𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
119 fvoveq1 7454 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉)))) = (𝐻‘(𝑧 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
120119cbvmptv 5261 . . . . . . . . . . 11 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) = (𝑧 ∈ (1...𝑊) ↦ (𝐻‘(𝑧 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
12142, 68, 41, 73, 43, 118, 120vdwlem2 17016 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝐾 + 1) MonoAP (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) → (𝐾 + 1) MonoAP 𝐻))
12265, 121sylbird 260 . . . . . . . . 9 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝐾 + 1) MonoAP 𝑔 → (𝐾 + 1) MonoAP 𝐻))
123122orim2d 968 . . . . . . . 8 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝑔) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
12454, 123syld 47 . . . . . . 7 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((⟨𝑀, 𝐾⟩ PolyAP 𝑔 ∨ (𝐾 + 1) MonoAP 𝑔) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
12539, 124mpd 15 . . . . . 6 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻))
126125expr 456 . . . . 5 ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
127126rexlimdvva 3211 . . . 4 (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
128127exlimdv 1931 . . 3 (𝜑 → (∃𝑔𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
12918, 128sylbid 240 . 2 (𝜑 → (𝐾 MonoAP 𝐹 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
13013, 129mpd 15 1 (𝜑 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963  {csn 4631  cop 4637   class class class wbr 5148  cmpt 5231  ccnv 5688  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  Fincfn 8984  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158   < clt 11293  cle 11294  cmin 11490  cn 12264  2c2 12319  0cn0 12524  cz 12611  cuz 12876  ...cfz 13544  APcvdwa 16999   MonoAP cvdwm 17000   PolyAP cvdwp 17001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-hash 14367  df-vdwap 17002  df-vdwmc 17003  df-vdwpc 17004
This theorem is referenced by:  vdwlem10  17024
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