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Theorem vdwlem9 16065
Description: Lemma for vdw 16070. (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 4878 . . 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 16060 . . . 4 (𝜑𝐹:(1...𝑉)⟶(𝑅𝑚 (1...𝑊)))
9 ovex 6938 . . . . 5 (𝑅𝑚 (1...𝑊)) ∈ V
10 ovex 6938 . . . . 5 (1...𝑉) ∈ V
119, 10elmap 8152 . . . 4 (𝐹 ∈ ((𝑅𝑚 (1...𝑊)) ↑𝑚 (1...𝑉)) ↔ 𝐹:(1...𝑉)⟶(𝑅𝑚 (1...𝑊)))
128, 11sylibr 226 . . 3 (𝜑𝐹 ∈ ((𝑅𝑚 (1...𝑊)) ↑𝑚 (1...𝑉)))
131, 2, 12rspcdva 3533 . 2 (𝜑𝐾 MonoAP 𝐹)
14 vdwlem9.k . . . . . 6 (𝜑𝐾 ∈ (ℤ‘2))
15 eluz2nn 12009 . . . . . 6 (𝐾 ∈ (ℤ‘2) → 𝐾 ∈ ℕ)
1614, 15syl 17 . . . . 5 (𝜑𝐾 ∈ ℕ)
1716nnnn0d 11679 . . . 4 (𝜑𝐾 ∈ ℕ0)
1810, 17, 8vdwmc 16054 . . 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 474 . . . . . . . 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 791 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))
2216adantr 474 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐾 ∈ ℕ)
23 simprll 799 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ ℕ)
24 simprlr 800 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑑 ∈ ℕ)
25 vdwapid1 16051 . . . . . . . . . . . . 13 ((𝐾 ∈ ℕ ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → 𝑎 ∈ (𝑎(AP‘𝐾)𝑑))
2622, 23, 24, 25syl3anc 1496 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ (𝑎(AP‘𝐾)𝑑))
2721, 26sseldd 3829 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ (𝐹 “ {𝑔}))
288ffnd 6280 . . . . . . . . . . . . 13 (𝜑𝐹 Fn (1...𝑉))
2928adantr 474 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐹 Fn (1...𝑉))
30 fniniseg 6588 . . . . . . . . . . . 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 224 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 ∈ (1...𝑉) ∧ (𝐹𝑎) = 𝑔))
3332simprd 491 . . . . . . . . 9 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝐹𝑎) = 𝑔)
348adantr 474 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐹:(1...𝑉)⟶(𝑅𝑚 (1...𝑊)))
3532simpld 490 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ (1...𝑉))
3634, 35ffvelrnd 6610 . . . . . . . . 9 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝐹𝑎) ∈ (𝑅𝑚 (1...𝑊)))
3733, 36eqeltrrd 2908 . . . . . . . 8 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑔 ∈ (𝑅𝑚 (1...𝑊)))
38 rsp 3139 . . . . . . . 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 474 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑉 ∈ ℕ)
414adantr 474 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑊 ∈ ℕ)
425adantr 474 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑅 ∈ Fin)
436adantr 474 . . . . . . . . . 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 474 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑀 ∈ ℕ)
46 ovex 6938 . . . . . . . . . . . 12 (1...𝑊) ∈ V
47 elmapg 8136 . . . . . . . . . . . 12 ((𝑅 ∈ Fin ∧ (1...𝑊) ∈ V) → (𝑔 ∈ (𝑅𝑚 (1...𝑊)) ↔ 𝑔:(1...𝑊)⟶𝑅))
4842, 46, 47sylancl 582 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑔 ∈ (𝑅𝑚 (1...𝑊)) ↔ 𝑔:(1...𝑊)⟶𝑅))
4937, 48mpbid 224 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑔:(1...𝑊)⟶𝑅)
5014adantr 474 . . . . . . . . . 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 16063 . . . . . . . . 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 901 . . . . . . . . . 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 892 . . . . . . . 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 6913 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥 − 1) = (𝑎 − 1))
5655oveq1d 6921 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥 − 1) + 𝑉) = ((𝑎 − 1) + 𝑉))
5756oveq2d 6922 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (𝑊 · ((𝑥 − 1) + 𝑉)) = (𝑊 · ((𝑎 − 1) + 𝑉)))
5857oveq2d 6922 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))) = (𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))
5958fveq2d 6438 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
6059mpteq2dv 4969 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))))
6146mptex 6743 . . . . . . . . . . . . . 14 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) ∈ V
6260, 7, 61fvmpt 6530 . . . . . . . . . . . . 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 2864 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) = 𝑔)
6564breq2d 4886 . . . . . . . . . 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 474 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐾 ∈ ℕ0)
67 peano2nn0 11661 . . . . . . . . . . . 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 11662 . . . . . . . . . . . . . 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 11652 . . . . . . . . . . . . 13 (((𝑎 − 1) ∈ ℕ0𝑉 ∈ ℕ) → ((𝑎 − 1) + 𝑉) ∈ ℕ)
7270, 40, 71syl2anc 581 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝑎 − 1) + 𝑉) ∈ ℕ)
7341, 72nnmulcld 11405 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · ((𝑎 − 1) + 𝑉)) ∈ ℕ)
7423, 40nnaddcld 11404 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ∈ ℕ)
7541, 74nnmulcld 11405 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ∈ ℕ)
7675nnzd 11810 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ∈ ℤ)
77 2nn 11425 . . . . . . . . . . . . . . . . 17 2 ∈ ℕ
78 nnmulcl 11376 . . . . . . . . . . . . . . . . 17 ((2 ∈ ℕ ∧ 𝑉 ∈ ℕ) → (2 · 𝑉) ∈ ℕ)
7977, 3, 78sylancr 583 . . . . . . . . . . . . . . . 16 (𝜑 → (2 · 𝑉) ∈ ℕ)
804, 79nnmulcld 11405 . . . . . . . . . . . . . . 15 (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℕ)
8180nnzd 11810 . . . . . . . . . . . . . 14 (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℤ)
8281adantr 474 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ ℤ)
8323nnred 11368 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ ℝ)
8440nnred 11368 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑉 ∈ ℝ)
85 elfzle2 12639 . . . . . . . . . . . . . . . . 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 10967 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ≤ (𝑉 + 𝑉))
8840nncnd 11369 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑉 ∈ ℂ)
89882timesd 11602 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (2 · 𝑉) = (𝑉 + 𝑉))
9087, 89breqtrrd 4902 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ≤ (2 · 𝑉))
9174nnred 11368 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ∈ ℝ)
9279nnred 11368 . . . . . . . . . . . . . . . 16 (𝜑 → (2 · 𝑉) ∈ ℝ)
9392adantr 474 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (2 · 𝑉) ∈ ℝ)
9441nnred 11368 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑊 ∈ ℝ)
9541nngt0d 11401 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 0 < 𝑊)
96 lemul2 11207 . . . . . . . . . . . . . . 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 1499 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝑎 + 𝑉) ≤ (2 · 𝑉) ↔ (𝑊 · (𝑎 + 𝑉)) ≤ (𝑊 · (2 · 𝑉))))
9890, 97mpbid 224 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ≤ (𝑊 · (2 · 𝑉)))
99 eluz2 11975 . . . . . . . . . . . . 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 1449 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ (ℤ‘(𝑊 · (𝑎 + 𝑉))))
10141nncnd 11369 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑊 ∈ ℂ)
102 1cnd 10352 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 1 ∈ ℂ)
10370nn0cnd 11681 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 − 1) ∈ ℂ)
104103, 88addcld 10377 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝑎 − 1) + 𝑉) ∈ ℂ)
105101, 102, 104adddid 10382 . . . . . . . . . . . . . 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 10380 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((1 + (𝑎 − 1)) + 𝑉) = (1 + ((𝑎 − 1) + 𝑉)))
107 ax-1cn 10311 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
10823nncnd 11369 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ ℂ)
109 pncan3 10610 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (1 + (𝑎 − 1)) = 𝑎)
110107, 108, 109sylancr 583 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (1 + (𝑎 − 1)) = 𝑎)
111110oveq1d 6921 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((1 + (𝑎 − 1)) + 𝑉) = (𝑎 + 𝑉))
112106, 111eqtr3d 2864 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (1 + ((𝑎 − 1) + 𝑉)) = (𝑎 + 𝑉))
113112oveq2d 6922 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (1 + ((𝑎 − 1) + 𝑉))) = (𝑊 · (𝑎 + 𝑉)))
114101mulid1d 10375 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · 1) = 𝑊)
115114oveq1d 6921 . . . . . . . . . . . . . 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 2870 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) = (𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉))))
117116fveq2d 6438 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (ℤ‘(𝑊 · (𝑎 + 𝑉))) = (ℤ‘(𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
118100, 117eleqtrd 2909 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ (ℤ‘(𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
119 fvoveq1 6929 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉)))) = (𝐻‘(𝑧 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
120119cbvmptv 4974 . . . . . . . . . . 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 16058 . . . . . . . . . 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 252 . . . . . . . . 9 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝐾 + 1) MonoAP 𝑔 → (𝐾 + 1) MonoAP 𝐻))
123122orim2d 996 . . . . . . . 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 450 . . . . 5 ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
127126rexlimdvva 3249 . . . 4 (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
128127exlimdv 2034 . . 3 (𝜑 → (∃𝑔𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
12918, 128sylbid 232 . 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 198  wa 386  wo 880   = wceq 1658  wex 1880  wcel 2166  wral 3118  wrex 3119  Vcvv 3415  wss 3799  {csn 4398  cop 4404   class class class wbr 4874  cmpt 4953  ccnv 5342  cima 5346   Fn wfn 6119  wf 6120  cfv 6124  (class class class)co 6906  𝑚 cmap 8123  Fincfn 8223  cc 10251  cr 10252  0cc0 10253  1c1 10254   + caddc 10256   · cmul 10258   < clt 10392  cle 10393  cmin 10586  cn 11351  2c2 11407  0cn0 11619  cz 11705  cuz 11969  ...cfz 12620  APcvdwa 16041   MonoAP cvdwm 16042   PolyAP cvdwp 16043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oadd 7831  df-er 8010  df-map 8125  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-card 9079  df-cda 9306  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-nn 11352  df-2 11415  df-n0 11620  df-z 11706  df-uz 11970  df-rp 12114  df-fz 12621  df-hash 13412  df-vdwap 16044  df-vdwmc 16045  df-vdwpc 16046
This theorem is referenced by:  vdwlem10  16066
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