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Theorem vdwlem9 16320
Description: Lemma for vdw 16325. (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 5067 . . 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 16315 . . . 4 (𝜑𝐹:(1...𝑉)⟶(𝑅m (1...𝑊)))
9 ovex 7183 . . . . 5 (𝑅m (1...𝑊)) ∈ V
10 ovex 7183 . . . . 5 (1...𝑉) ∈ V
119, 10elmap 8430 . . . 4 (𝐹 ∈ ((𝑅m (1...𝑊)) ↑m (1...𝑉)) ↔ 𝐹:(1...𝑉)⟶(𝑅m (1...𝑊)))
128, 11sylibr 235 . . 3 (𝜑𝐹 ∈ ((𝑅m (1...𝑊)) ↑m (1...𝑉)))
131, 2, 12rspcdva 3629 . 2 (𝜑𝐾 MonoAP 𝐹)
14 vdwlem9.k . . . . . 6 (𝜑𝐾 ∈ (ℤ‘2))
15 eluz2nn 12278 . . . . . 6 (𝐾 ∈ (ℤ‘2) → 𝐾 ∈ ℕ)
1614, 15syl 17 . . . . 5 (𝜑𝐾 ∈ ℕ)
1716nnnn0d 11949 . . . 4 (𝜑𝐾 ∈ ℕ0)
1810, 17, 8vdwmc 16309 . . 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 481 . . . . . . . 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 769 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))
2216adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐾 ∈ ℕ)
23 simprll 775 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ ℕ)
24 simprlr 776 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑑 ∈ ℕ)
25 vdwapid1 16306 . . . . . . . . . . . . 13 ((𝐾 ∈ ℕ ∧ 𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) → 𝑎 ∈ (𝑎(AP‘𝐾)𝑑))
2622, 23, 24, 25syl3anc 1365 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ (𝑎(AP‘𝐾)𝑑))
2721, 26sseldd 3972 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ (𝐹 “ {𝑔}))
288ffnd 6514 . . . . . . . . . . . . 13 (𝜑𝐹 Fn (1...𝑉))
2928adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐹 Fn (1...𝑉))
30 fniniseg 6828 . . . . . . . . . . . 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 233 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 ∈ (1...𝑉) ∧ (𝐹𝑎) = 𝑔))
3332simprd 496 . . . . . . . . 9 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝐹𝑎) = 𝑔)
348adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐹:(1...𝑉)⟶(𝑅m (1...𝑊)))
3532simpld 495 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ (1...𝑉))
3634, 35ffvelrnd 6850 . . . . . . . . 9 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝐹𝑎) ∈ (𝑅m (1...𝑊)))
3733, 36eqeltrrd 2919 . . . . . . . 8 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑔 ∈ (𝑅m (1...𝑊)))
38 rsp 3210 . . . . . . . 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 481 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑉 ∈ ℕ)
414adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑊 ∈ ℕ)
425adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑅 ∈ Fin)
436adantr 481 . . . . . . . . . 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 481 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑀 ∈ ℕ)
46 ovex 7183 . . . . . . . . . . . 12 (1...𝑊) ∈ V
47 elmapg 8414 . . . . . . . . . . . 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 233 . . . . . . . . . 10 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑔:(1...𝑊)⟶𝑅)
5014adantr 481 . . . . . . . . . 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 16318 . . . . . . . . 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 864 . . . . . . . . . 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 855 . . . . . . . 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 7157 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝑥 − 1) = (𝑎 − 1))
5655oveq1d 7165 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝑥 − 1) + 𝑉) = ((𝑎 − 1) + 𝑉))
5756oveq2d 7166 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (𝑊 · ((𝑥 − 1) + 𝑉)) = (𝑊 · ((𝑎 − 1) + 𝑉)))
5857oveq2d 7166 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))) = (𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))
5958fveq2d 6673 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉)))) = (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
6059mpteq2dv 5159 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))) = (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))))
6146mptex 6983 . . . . . . . . . . . . . 14 (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) ∈ V
6260, 7, 61fvmpt 6767 . . . . . . . . . . . . 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 2863 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉))))) = 𝑔)
6564breq2d 5075 . . . . . . . . . 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 481 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝐾 ∈ ℕ0)
67 peano2nn0 11931 . . . . . . . . . . . 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 11932 . . . . . . . . . . . . . 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 11922 . . . . . . . . . . . . 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 11684 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · ((𝑎 − 1) + 𝑉)) ∈ ℕ)
7423, 40nnaddcld 11683 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ∈ ℕ)
7541, 74nnmulcld 11684 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ∈ ℕ)
7675nnzd 12080 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ∈ ℤ)
77 2nn 11704 . . . . . . . . . . . . . . . . 17 2 ∈ ℕ
78 nnmulcl 11655 . . . . . . . . . . . . . . . . 17 ((2 ∈ ℕ ∧ 𝑉 ∈ ℕ) → (2 · 𝑉) ∈ ℕ)
7977, 3, 78sylancr 587 . . . . . . . . . . . . . . . 16 (𝜑 → (2 · 𝑉) ∈ ℕ)
804, 79nnmulcld 11684 . . . . . . . . . . . . . . 15 (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℕ)
8180nnzd 12080 . . . . . . . . . . . . . 14 (𝜑 → (𝑊 · (2 · 𝑉)) ∈ ℤ)
8281adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ ℤ)
8323nnred 11647 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ ℝ)
8440nnred 11647 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑉 ∈ ℝ)
85 elfzle2 12906 . . . . . . . . . . . . . . . . 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 11248 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ≤ (𝑉 + 𝑉))
8840nncnd 11648 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑉 ∈ ℂ)
89882timesd 11874 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (2 · 𝑉) = (𝑉 + 𝑉))
9087, 89breqtrrd 5091 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ≤ (2 · 𝑉))
9174nnred 11647 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 + 𝑉) ∈ ℝ)
9279nnred 11647 . . . . . . . . . . . . . . . 16 (𝜑 → (2 · 𝑉) ∈ ℝ)
9392adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (2 · 𝑉) ∈ ℝ)
9441nnred 11647 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑊 ∈ ℝ)
9541nngt0d 11680 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 0 < 𝑊)
96 lemul2 11487 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝑎 + 𝑉) ≤ (2 · 𝑉) ↔ (𝑊 · (𝑎 + 𝑉)) ≤ (𝑊 · (2 · 𝑉))))
9890, 97mpbid 233 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) ≤ (𝑊 · (2 · 𝑉)))
99 eluz2 12243 . . . . . . . . . . . . 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 1337 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ (ℤ‘(𝑊 · (𝑎 + 𝑉))))
10141nncnd 11648 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑊 ∈ ℂ)
102 1cnd 10630 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 1 ∈ ℂ)
10370nn0cnd 11951 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑎 − 1) ∈ ℂ)
104103, 88addcld 10654 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝑎 − 1) + 𝑉) ∈ ℂ)
105101, 102, 104adddid 10659 . . . . . . . . . . . . . 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 10657 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((1 + (𝑎 − 1)) + 𝑉) = (1 + ((𝑎 − 1) + 𝑉)))
107 ax-1cn 10589 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
10823nncnd 11648 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → 𝑎 ∈ ℂ)
109 pncan3 10888 . . . . . . . . . . . . . . . . . 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 7165 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((1 + (𝑎 − 1)) + 𝑉) = (𝑎 + 𝑉))
112106, 111eqtr3d 2863 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (1 + ((𝑎 − 1) + 𝑉)) = (𝑎 + 𝑉))
113112oveq2d 7166 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (1 + ((𝑎 − 1) + 𝑉))) = (𝑊 · (𝑎 + 𝑉)))
114101mulid1d 10652 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · 1) = 𝑊)
115114oveq1d 7165 . . . . . . . . . . . . . 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 2869 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (𝑎 + 𝑉)) = (𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉))))
117116fveq2d 6673 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (ℤ‘(𝑊 · (𝑎 + 𝑉))) = (ℤ‘(𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
118100, 117eleqtrd 2920 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → (𝑊 · (2 · 𝑉)) ∈ (ℤ‘(𝑊 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
119 fvoveq1 7173 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝐻‘(𝑦 + (𝑊 · ((𝑎 − 1) + 𝑉)))) = (𝐻‘(𝑧 + (𝑊 · ((𝑎 − 1) + 𝑉)))))
120119cbvmptv 5166 . . . . . . . . . . 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 16313 . . . . . . . . . 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 261 . . . . . . . . 9 ((𝜑 ∧ ((𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ) ∧ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}))) → ((𝐾 + 1) MonoAP 𝑔 → (𝐾 + 1) MonoAP 𝐻))
123122orim2d 962 . . . . . . . 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 457 . . . . 5 ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ ℕ)) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
127126rexlimdvva 3299 . . . 4 (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
128127exlimdv 1927 . . 3 (𝜑 → (∃𝑔𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝐹 “ {𝑔}) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐻)))
12918, 128sylbid 241 . 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 207  wa 396  wo 843   = wceq 1530  wex 1773  wcel 2107  wral 3143  wrex 3144  Vcvv 3500  wss 3940  {csn 4564  cop 4570   class class class wbr 5063  cmpt 5143  ccnv 5553  cima 5557   Fn wfn 6349  wf 6350  cfv 6354  (class class class)co 7150  m cmap 8401  Fincfn 8503  cc 10529  cr 10530  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536   < clt 10669  cle 10670  cmin 10864  cn 11632  2c2 11686  0cn0 11891  cz 11975  cuz 12237  ...cfz 12887  APcvdwa 16296   MonoAP cvdwm 16297   PolyAP cvdwp 16298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12385  df-fz 12888  df-hash 13686  df-vdwap 16299  df-vdwmc 16300  df-vdwpc 16301
This theorem is referenced by:  vdwlem10  16321
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