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Theorem vdwlem13 16865
Description: Lemma for vdw 16866. Main induction on 𝐾; 𝐾 = 0, 𝐾 = 1 base cases. (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
vdw.r (𝜑𝑅 ∈ Fin)
vdw.k (𝜑𝐾 ∈ ℕ0)
Assertion
Ref Expression
vdwlem13 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓)
Distinct variable groups:   𝜑,𝑛,𝑓   𝑓,𝐾,𝑛   𝑅,𝑓,𝑛   𝜑,𝑓

Proof of Theorem vdwlem13
Dummy variables 𝑎 𝑐 𝑑 𝑔 𝑘 𝑚 𝑥 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn1uz2 12850 . . 3 (𝐾 ∈ ℕ ↔ (𝐾 = 1 ∨ 𝐾 ∈ (ℤ‘2)))
2 vdw.r . . . . . . . . . 10 (𝜑𝑅 ∈ Fin)
3 ovex 7390 . . . . . . . . . 10 (1...1) ∈ V
4 elmapg 8778 . . . . . . . . . 10 ((𝑅 ∈ Fin ∧ (1...1) ∈ V) → (𝑓 ∈ (𝑅m (1...1)) ↔ 𝑓:(1...1)⟶𝑅))
52, 3, 4sylancl 586 . . . . . . . . 9 (𝜑 → (𝑓 ∈ (𝑅m (1...1)) ↔ 𝑓:(1...1)⟶𝑅))
65biimpa 477 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑅m (1...1))) → 𝑓:(1...1)⟶𝑅)
7 1nn 12164 . . . . . . . . . 10 1 ∈ ℕ
8 vdwap1 16849 . . . . . . . . . 10 ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘1)1) = {1})
97, 7, 8mp2an 690 . . . . . . . . 9 (1(AP‘1)1) = {1}
10 1z 12533 . . . . . . . . . . . 12 1 ∈ ℤ
11 elfz3 13451 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ (1...1))
1210, 11mp1i 13 . . . . . . . . . . 11 ((𝜑𝑓:(1...1)⟶𝑅) → 1 ∈ (1...1))
13 eqidd 2737 . . . . . . . . . . 11 ((𝜑𝑓:(1...1)⟶𝑅) → (𝑓‘1) = (𝑓‘1))
14 ffn 6668 . . . . . . . . . . . . 13 (𝑓:(1...1)⟶𝑅𝑓 Fn (1...1))
1514adantl 482 . . . . . . . . . . . 12 ((𝜑𝑓:(1...1)⟶𝑅) → 𝑓 Fn (1...1))
16 fniniseg 7010 . . . . . . . . . . . 12 (𝑓 Fn (1...1) → (1 ∈ (𝑓 “ {(𝑓‘1)}) ↔ (1 ∈ (1...1) ∧ (𝑓‘1) = (𝑓‘1))))
1715, 16syl 17 . . . . . . . . . . 11 ((𝜑𝑓:(1...1)⟶𝑅) → (1 ∈ (𝑓 “ {(𝑓‘1)}) ↔ (1 ∈ (1...1) ∧ (𝑓‘1) = (𝑓‘1))))
1812, 13, 17mpbir2and 711 . . . . . . . . . 10 ((𝜑𝑓:(1...1)⟶𝑅) → 1 ∈ (𝑓 “ {(𝑓‘1)}))
1918snssd 4769 . . . . . . . . 9 ((𝜑𝑓:(1...1)⟶𝑅) → {1} ⊆ (𝑓 “ {(𝑓‘1)}))
209, 19eqsstrid 3992 . . . . . . . 8 ((𝜑𝑓:(1...1)⟶𝑅) → (1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)}))
216, 20syldan 591 . . . . . . 7 ((𝜑𝑓 ∈ (𝑅m (1...1))) → (1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)}))
2221ralrimiva 3143 . . . . . 6 (𝜑 → ∀𝑓 ∈ (𝑅m (1...1))(1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)}))
23 fveq2 6842 . . . . . . . . 9 (𝐾 = 1 → (AP‘𝐾) = (AP‘1))
2423oveqd 7374 . . . . . . . 8 (𝐾 = 1 → (1(AP‘𝐾)1) = (1(AP‘1)1))
2524sseq1d 3975 . . . . . . 7 (𝐾 = 1 → ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)})))
2625ralbidv 3174 . . . . . 6 (𝐾 = 1 → (∀𝑓 ∈ (𝑅m (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ ∀𝑓 ∈ (𝑅m (1...1))(1(AP‘1)1) ⊆ (𝑓 “ {(𝑓‘1)})))
2722, 26syl5ibrcom 246 . . . . 5 (𝜑 → (𝐾 = 1 → ∀𝑓 ∈ (𝑅m (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)})))
28 oveq1 7364 . . . . . . . . . . . 12 (𝑎 = 1 → (𝑎(AP‘𝐾)𝑑) = (1(AP‘𝐾)𝑑))
2928sseq1d 3975 . . . . . . . . . . 11 (𝑎 = 1 → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)})))
30 oveq2 7365 . . . . . . . . . . . 12 (𝑑 = 1 → (1(AP‘𝐾)𝑑) = (1(AP‘𝐾)1))
3130sseq1d 3975 . . . . . . . . . . 11 (𝑑 = 1 → ((1(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)})))
3229, 31rspc2ev 3592 . . . . . . . . . 10 ((1 ∈ ℕ ∧ 1 ∈ ℕ ∧ (1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)})) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}))
337, 7, 32mp3an12 1451 . . . . . . . . 9 ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}))
34 fvex 6855 . . . . . . . . . 10 (𝑓‘1) ∈ V
35 sneq 4596 . . . . . . . . . . . . 13 (𝑐 = (𝑓‘1) → {𝑐} = {(𝑓‘1)})
3635imaeq2d 6013 . . . . . . . . . . . 12 (𝑐 = (𝑓‘1) → (𝑓 “ {𝑐}) = (𝑓 “ {(𝑓‘1)}))
3736sseq2d 3976 . . . . . . . . . . 11 (𝑐 = (𝑓‘1) → ((𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)})))
38372rexbidv 3213 . . . . . . . . . 10 (𝑐 = (𝑓‘1) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)})))
3934, 38spcev 3565 . . . . . . . . 9 (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {(𝑓‘1)}) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐}))
4033, 39syl 17 . . . . . . . 8 ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐}))
41 vdw.k . . . . . . . . . 10 (𝜑𝐾 ∈ ℕ0)
4241adantr 481 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝑅m (1...1))) → 𝐾 ∈ ℕ0)
433, 42, 6vdwmc 16850 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑅m (1...1))) → (𝐾 MonoAP 𝑓 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (𝑓 “ {𝑐})))
4440, 43syl5ibr 245 . . . . . . 7 ((𝜑𝑓 ∈ (𝑅m (1...1))) → ((1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → 𝐾 MonoAP 𝑓))
4544ralimdva 3164 . . . . . 6 (𝜑 → (∀𝑓 ∈ (𝑅m (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∀𝑓 ∈ (𝑅m (1...1))𝐾 MonoAP 𝑓))
46 oveq2 7365 . . . . . . . . . 10 (𝑛 = 1 → (1...𝑛) = (1...1))
4746oveq2d 7373 . . . . . . . . 9 (𝑛 = 1 → (𝑅m (1...𝑛)) = (𝑅m (1...1)))
4847raleqdv 3313 . . . . . . . 8 (𝑛 = 1 → (∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅m (1...1))𝐾 MonoAP 𝑓))
4948rspcev 3581 . . . . . . 7 ((1 ∈ ℕ ∧ ∀𝑓 ∈ (𝑅m (1...1))𝐾 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓)
507, 49mpan 688 . . . . . 6 (∀𝑓 ∈ (𝑅m (1...1))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓)
5145, 50syl6 35 . . . . 5 (𝜑 → (∀𝑓 ∈ (𝑅m (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
5227, 51syld 47 . . . 4 (𝜑 → (𝐾 = 1 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
53 breq1 5108 . . . . . . . 8 (𝑥 = 2 → (𝑥 MonoAP 𝑓 ↔ 2 MonoAP 𝑓))
5453rexralbidv 3214 . . . . . . 7 (𝑥 = 2 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))2 MonoAP 𝑓))
5554ralbidv 3174 . . . . . 6 (𝑥 = 2 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))2 MonoAP 𝑓))
56 breq1 5108 . . . . . . . 8 (𝑥 = 𝑘 → (𝑥 MonoAP 𝑓𝑘 MonoAP 𝑓))
5756rexralbidv 3214 . . . . . . 7 (𝑥 = 𝑘 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓))
5857ralbidv 3174 . . . . . 6 (𝑥 = 𝑘 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓))
59 breq1 5108 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (𝑥 MonoAP 𝑓 ↔ (𝑘 + 1) MonoAP 𝑓))
6059rexralbidv 3214 . . . . . . 7 (𝑥 = (𝑘 + 1) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
6160ralbidv 3174 . . . . . 6 (𝑥 = (𝑘 + 1) → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
62 breq1 5108 . . . . . . . 8 (𝑥 = 𝐾 → (𝑥 MonoAP 𝑓𝐾 MonoAP 𝑓))
6362rexralbidv 3214 . . . . . . 7 (𝑥 = 𝐾 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝐾 MonoAP 𝑓))
6463ralbidv 3174 . . . . . 6 (𝑥 = 𝐾 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝐾 MonoAP 𝑓))
65 hashcl 14256 . . . . . . . . 9 (𝑟 ∈ Fin → (♯‘𝑟) ∈ ℕ0)
66 nn0p1nn 12452 . . . . . . . . 9 ((♯‘𝑟) ∈ ℕ0 → ((♯‘𝑟) + 1) ∈ ℕ)
6765, 66syl 17 . . . . . . . 8 (𝑟 ∈ Fin → ((♯‘𝑟) + 1) ∈ ℕ)
68 simpll 765 . . . . . . . . . . 11 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑟 ∈ Fin)
69 simplr 767 . . . . . . . . . . . 12 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1))))
70 vex 3449 . . . . . . . . . . . . 13 𝑟 ∈ V
71 ovex 7390 . . . . . . . . . . . . 13 (1...((♯‘𝑟) + 1)) ∈ V
7270, 71elmap 8809 . . . . . . . . . . . 12 (𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1))) ↔ 𝑓:(1...((♯‘𝑟) + 1))⟶𝑟)
7369, 72sylib 217 . . . . . . . . . . 11 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓:(1...((♯‘𝑟) + 1))⟶𝑟)
74 simpr 485 . . . . . . . . . . 11 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → ¬ 2 MonoAP 𝑓)
7568, 73, 74vdwlem12 16864 . . . . . . . . . 10 ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓)
76 iman 402 . . . . . . . . . 10 (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) → 2 MonoAP 𝑓) ↔ ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓))
7775, 76mpbir 230 . . . . . . . . 9 ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))) → 2 MonoAP 𝑓)
7877ralrimiva 3143 . . . . . . . 8 (𝑟 ∈ Fin → ∀𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓)
79 oveq2 7365 . . . . . . . . . . 11 (𝑛 = ((♯‘𝑟) + 1) → (1...𝑛) = (1...((♯‘𝑟) + 1)))
8079oveq2d 7373 . . . . . . . . . 10 (𝑛 = ((♯‘𝑟) + 1) → (𝑟m (1...𝑛)) = (𝑟m (1...((♯‘𝑟) + 1))))
8180raleqdv 3313 . . . . . . . . 9 (𝑛 = ((♯‘𝑟) + 1) → (∀𝑓 ∈ (𝑟m (1...𝑛))2 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓))
8281rspcev 3581 . . . . . . . 8 ((((♯‘𝑟) + 1) ∈ ℕ ∧ ∀𝑓 ∈ (𝑟m (1...((♯‘𝑟) + 1)))2 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))2 MonoAP 𝑓)
8367, 78, 82syl2anc 584 . . . . . . 7 (𝑟 ∈ Fin → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))2 MonoAP 𝑓)
8483rgen 3066 . . . . . 6 𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))2 MonoAP 𝑓
85 oveq1 7364 . . . . . . . . . . 11 (𝑟 = 𝑠 → (𝑟m (1...𝑛)) = (𝑠m (1...𝑛)))
8685raleqdv 3313 . . . . . . . . . 10 (𝑟 = 𝑠 → (∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓))
8786rexbidv 3175 . . . . . . . . 9 (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓))
88 oveq2 7365 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
8988oveq2d 7373 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝑠m (1...𝑛)) = (𝑠m (1...𝑚)))
9089raleqdv 3313 . . . . . . . . . . 11 (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑓))
91 breq2 5109 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑘 MonoAP 𝑓𝑘 MonoAP 𝑔))
9291cbvralvw 3225 . . . . . . . . . . 11 (∀𝑓 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔)
9390, 92bitrdi 286 . . . . . . . . . 10 (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔))
9493cbvrexvw 3226 . . . . . . . . 9 (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔)
9587, 94bitrdi 286 . . . . . . . 8 (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔))
9695cbvralvw 3225 . . . . . . 7 (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔)
97 simplr 767 . . . . . . . . . 10 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔) → 𝑟 ∈ Fin)
98 simpll 765 . . . . . . . . . 10 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔) → 𝑘 ∈ (ℤ‘2))
99 simpr 485 . . . . . . . . . . 11 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔) → ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔)
10094ralbii 3096 . . . . . . . . . . 11 (∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔)
10199, 100sylibr 233 . . . . . . . . . 10 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔) → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠m (1...𝑛))𝑘 MonoAP 𝑓)
10297, 98, 101vdwlem11 16863 . . . . . . . . 9 (((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))(𝑘 + 1) MonoAP 𝑓)
103102ex 413 . . . . . . . 8 ((𝑘 ∈ (ℤ‘2) ∧ 𝑟 ∈ Fin) → (∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
104103ralrimdva 3151 . . . . . . 7 (𝑘 ∈ (ℤ‘2) → (∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠m (1...𝑚))𝑘 MonoAP 𝑔 → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
10596, 104biimtrid 241 . . . . . 6 (𝑘 ∈ (ℤ‘2) → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝑘 MonoAP 𝑓 → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
10655, 58, 61, 64, 84, 105uzind4i 12835 . . . . 5 (𝐾 ∈ (ℤ‘2) → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝐾 MonoAP 𝑓)
107 oveq1 7364 . . . . . . . 8 (𝑟 = 𝑅 → (𝑟m (1...𝑛)) = (𝑅m (1...𝑛)))
108107raleqdv 3313 . . . . . . 7 (𝑟 = 𝑅 → (∀𝑓 ∈ (𝑟m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
109108rexbidv 3175 . . . . . 6 (𝑟 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
110109rspcv 3577 . . . . 5 (𝑅 ∈ Fin → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟m (1...𝑛))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
1112, 106, 110syl2im 40 . . . 4 (𝜑 → (𝐾 ∈ (ℤ‘2) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
11252, 111jaod 857 . . 3 (𝜑 → ((𝐾 = 1 ∨ 𝐾 ∈ (ℤ‘2)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
1131, 112biimtrid 241 . 2 (𝜑 → (𝐾 ∈ ℕ → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
114 fveq2 6842 . . . . . . 7 (𝐾 = 0 → (AP‘𝐾) = (AP‘0))
115114oveqd 7374 . . . . . 6 (𝐾 = 0 → (1(AP‘𝐾)1) = (1(AP‘0)1))
116 vdwap0 16848 . . . . . . 7 ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘0)1) = ∅)
1177, 7, 116mp2an 690 . . . . . 6 (1(AP‘0)1) = ∅
118115, 117eqtrdi 2792 . . . . 5 (𝐾 = 0 → (1(AP‘𝐾)1) = ∅)
119 0ss 4356 . . . . 5 ∅ ⊆ (𝑓 “ {(𝑓‘1)})
120118, 119eqsstrdi 3998 . . . 4 (𝐾 = 0 → (1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}))
121120ralrimivw 3147 . . 3 (𝐾 = 0 → ∀𝑓 ∈ (𝑅m (1...1))(1(AP‘𝐾)1) ⊆ (𝑓 “ {(𝑓‘1)}))
122121, 51syl5 34 . 2 (𝜑 → (𝐾 = 0 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓))
123 elnn0 12415 . . 3 (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
12441, 123sylib 217 . 2 (𝜑 → (𝐾 ∈ ℕ ∨ 𝐾 = 0))
125113, 122, 124mpjaod 858 1 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅m (1...𝑛))𝐾 MonoAP 𝑓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910  c0 4282  {csn 4586   class class class wbr 5105  ccnv 5632  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  Fincfn 8883  0cc0 11051  1c1 11052   + caddc 11054  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  chash 14230  APcvdwa 16837   MonoAP cvdwm 16838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-hash 14231  df-vdwap 16840  df-vdwmc 16841  df-vdwpc 16842
This theorem is referenced by:  vdw  16866
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