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Theorem vdwlem9 17027
Description: Lemma for vdw 17032. (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 5147 . . 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 17022 . . . 4 (𝜑𝐹:(1...𝑉)⟶(𝑅m (1...𝑊)))
9 ovex 7464 . . . . 5 (𝑅m (1...𝑊)) ∈ V
10 ovex 7464 . . . . 5 (1...𝑉) ∈ V
119, 10elmap 8911 . . . 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 12924 . . . . . 6 (𝐾 ∈ (ℤ‘2) → 𝐾 ∈ ℕ)
1614, 15syl 17 . . . . 5 (𝜑𝐾 ∈ ℕ)
1716nnnn0d 12587 . . . 4 (𝜑𝐾 ∈ ℕ0)
1810, 17, 8vdwmc 17016 . . 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 17013 . . . . . . . . . . . . 13 ((𝐾 ∈ ℕ ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → 𝑎 ∈ (𝑎(AP‘𝐾)𝑑))
2622, 23, 24, 25syl3anc 1373 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ (𝑎(AP‘𝐾)𝑑))
2721, 26sseldd 3984 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ (𝐹 “ {𝑔}))
288ffnd 6737 . . . . . . . . . . . . 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 2842 . . . . . . . 8 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑔 ∈ (𝑅m (1...𝑊)))
38 rsp 3247 . . . . . . . 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 8879 . . . . . . . . . . . 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 17025 . . . . . . . . 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 869 . . . . . . . . . 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 860 . . . . . . . 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 6910 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
6059mpteq2dv 5244 . . . . . . . . . . . . . 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 2779 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) = 𝑔)
6564breq2d 5155 . . . . . . . . . 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 12566 . . . . . . . . . . . 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 12567 . . . . . . . . . . . . . 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 12557 . . . . . . . . . . . . 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 12319 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · ((𝑎 − 1) + 𝑉)) ∈ ℕ)
7423, 40nnaddcld 12318 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ∈ ℕ)
7541, 74nnmulcld 12319 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ∈ ℕ)
7675nnzd 12640 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ∈ ℤ)
77 2nn 12339 . . . . . . . . . . . . . . . . 17 2 ∈ ℕ
78 nnmulcl 12290 . . . . . . . . . . . . . . . . 17 ((2 ∈ ℕ ∧ 𝑉 ∈ ℕ) → (2 · 𝑉) ∈ ℕ)
7977, 3, 78sylancr 587 . . . . . . . . . . . . . . . 16 (𝜑 → (2 · 𝑉) ∈ ℕ)
804, 79nnmulcld 12319 . . . . . . . . . . . . . . 15 (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℕ)
8180nnzd 12640 . . . . . . . . . . . . . 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 12281 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ ℝ)
8440nnred 12281 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑉 ∈ ℝ)
85 elfzle2 13568 . . . . . . . . . . . . . . . . 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 11877 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ≤ (𝑉 + 𝑉))
8840nncnd 12282 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑉 ∈ ℂ)
89882timesd 12509 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (2 · 𝑉) = (𝑉 + 𝑉))
9087, 89breqtrrd 5171 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ≤ (2 · 𝑉))
9174nnred 12281 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ∈ ℝ)
9279nnred 12281 . . . . . . . . . . . . . . . 16 (𝜑 → (2 · 𝑉) ∈ ℝ)
9392adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (2 · 𝑉) ∈ ℝ)
9441nnred 12281 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑊 ∈ ℝ)
9541nngt0d 12315 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 0 < 𝑊)
96 lemul2 12120 . . . . . . . . . . . . . . 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 1376 . . . . . . . . . . . . . 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 12884 . . . . . . . . . . . . 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 1344 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ (ℤ‘(𝑊 · (𝑎 + 𝑉))))
10141nncnd 12282 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑊 ∈ ℂ)
102 1cnd 11256 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 1 ∈ ℂ)
10370nn0cnd 12589 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 − 1) ∈ ℂ)
104103, 88addcld 11280 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝑎 − 1) + 𝑉) ∈ ℂ)
105101, 102, 104adddid 11285 . . . . . . . . . . . . . 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 11283 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((1 + (𝑎 − 1)) + 𝑉) = (1 + ((𝑎 − 1) + 𝑉)))
107 ax-1cn 11213 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
10823nncnd 12282 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ ℂ)
109 pncan3 11516 . . . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . 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 11278 . . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) = (𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉))))
117116fveq2d 6910 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (ℤ‘(𝑊 · (𝑎 + 𝑉))) = (ℤ‘(𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
118100, 117eleqtrd 2843 . . . . . . . . . . 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 5255 . . . . . . . . . . 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 17020 . . . . . . . . . 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 969 . . . . . . . 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 3213 . . . 4 (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
128127exlimdv 1933 . . 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 848   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951  {csn 4626  cop 4632   class class class wbr 5143  cmpt 5225  ccnv 5684  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  Fincfn 8985  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295  cle 11296  cmin 11492  cn 12266  2c2 12321  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547  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:  vdwlem10  17028
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