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Theorem dyadmax 24126
Description: Any nonempty set of dyadic rational intervals has a maximal element. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypothesis
Ref Expression
dyadmbl.1 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
Assertion
Ref Expression
dyadmax ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))
Distinct variable groups:   𝑥,𝑦   𝑧,𝑤,𝑥,𝑦,𝐴   𝑤,𝐹,𝑥,𝑦,𝑧

Proof of Theorem dyadmax
Dummy variables 𝑐 𝑑 𝑎 𝑏 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltweuz 13317 . . . . 5 < We (ℤ‘0)
21a1i 11 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → < We (ℤ‘0))
3 nn0ex 11891 . . . . . 6 0 ∈ V
43rabex 5226 . . . . 5 {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V
54a1i 11 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V)
6 ssrab2 4053 . . . . . 6 {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ ℕ0
7 nn0uz 12268 . . . . . 6 0 = (ℤ‘0)
86, 7sseqtri 4000 . . . . 5 {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ (ℤ‘0)
98a1i 11 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ (ℤ‘0))
10 id 22 . . . . . . 7 (𝐴 ≠ ∅ → 𝐴 ≠ ∅)
11 dyadmbl.1 . . . . . . . . . . . 12 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
1211dyadf 24119 . . . . . . . . . . 11 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
13 ffn 6507 . . . . . . . . . . 11 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14 ovelrn 7313 . . . . . . . . . . 11 (𝐹 Fn (ℤ × ℕ0) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛)))
1512, 13, 14mp2b 10 . . . . . . . . . 10 (𝑧 ∈ ran 𝐹 ↔ ∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛))
16 rexcom 3352 . . . . . . . . . 10 (∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
1715, 16sylbb 220 . . . . . . . . 9 (𝑧 ∈ ran 𝐹 → ∃𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
1817rgen 3145 . . . . . . . 8 𝑧 ∈ ran 𝐹𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)
19 ssralv 4030 . . . . . . . 8 (𝐴 ⊆ ran 𝐹 → (∀𝑧 ∈ ran 𝐹𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) → ∀𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)))
2018, 19mpi 20 . . . . . . 7 (𝐴 ⊆ ran 𝐹 → ∀𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
21 r19.2z 4436 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)) → ∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2210, 20, 21syl2anr 596 . . . . . 6 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
23 rexcom 3352 . . . . . 6 (∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2422, 23sylib 219 . . . . 5 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑛 ∈ ℕ0𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
25 rabn0 4336 . . . . 5 ({𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅ ↔ ∃𝑛 ∈ ℕ0𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2624, 25sylibr 235 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅)
27 wereu 5544 . . . 4 (( < We (ℤ‘0) ∧ ({𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V ∧ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ (ℤ‘0) ∧ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅)) → ∃!𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
282, 5, 9, 26, 27syl13anc 1364 . . 3 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃!𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
29 reurex 3429 . . 3 (∃!𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
3028, 29syl 17 . 2 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
31 oveq2 7153 . . . . . . 7 (𝑛 = 𝑐 → (𝑎𝐹𝑛) = (𝑎𝐹𝑐))
3231eqeq2d 2829 . . . . . 6 (𝑛 = 𝑐 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑧 = (𝑎𝐹𝑐)))
33322rexbidv 3297 . . . . 5 (𝑛 = 𝑐 → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
3433elrab 3677 . . . 4 (𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ↔ (𝑐 ∈ ℕ0 ∧ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
35 eqeq1 2822 . . . . . . . . . 10 (𝑧 = 𝑤 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑎𝐹𝑛)))
36 oveq1 7152 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎𝐹𝑛) = (𝑏𝐹𝑛))
3736eqeq2d 2829 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑤 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑛)))
3835, 37cbvrex2vw 3460 . . . . . . . . 9 (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛))
39 oveq2 7153 . . . . . . . . . . 11 (𝑛 = 𝑑 → (𝑏𝐹𝑛) = (𝑏𝐹𝑑))
4039eqeq2d 2829 . . . . . . . . . 10 (𝑛 = 𝑑 → (𝑤 = (𝑏𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑑)))
41402rexbidv 3297 . . . . . . . . 9 (𝑛 = 𝑑 → (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
4238, 41syl5bb 284 . . . . . . . 8 (𝑛 = 𝑑 → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
4342ralrab 3682 . . . . . . 7 (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 ↔ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
44 r19.23v 3276 . . . . . . . . . . . . . . . . 17 (∀𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
4544ralbii 3162 . . . . . . . . . . . . . . . 16 (∀𝑑 ∈ ℕ0𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
46 ralcom 3351 . . . . . . . . . . . . . . . 16 (∀𝑑 ∈ ℕ0𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤𝐴𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
4745, 46bitr3i 278 . . . . . . . . . . . . . . 15 (∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤𝐴𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
48 simplll 771 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → 𝐴 ⊆ ran 𝐹)
4948sselda 3964 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → 𝑤 ∈ ran 𝐹)
50 ovelrn 7313 . . . . . . . . . . . . . . . . . . . 20 (𝐹 Fn (ℤ × ℕ0) → (𝑤 ∈ ran 𝐹 ↔ ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑)))
5112, 13, 50mp2b 10 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ran 𝐹 ↔ ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑))
5249, 51sylib 219 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑))
53 rexcom 3352 . . . . . . . . . . . . . . . . . . 19 (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) ↔ ∃𝑑 ∈ ℕ0𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))
54 r19.29 3251 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑑 ∈ ℕ0𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)) → ∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
5554expcom 414 . . . . . . . . . . . . . . . . . . 19 (∃𝑑 ∈ ℕ0𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → (∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))))
5653, 55sylbi 218 . . . . . . . . . . . . . . . . . 18 (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) → (∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))))
5752, 56syl 17 . . . . . . . . . . . . . . . . 17 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → (∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))))
58 simplrr 774 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → 𝑎 ∈ ℤ)
5958ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑎 ∈ ℤ)
60 simplrr 774 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑏 ∈ ℤ)
61 simp-5r 782 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑐 ∈ ℕ0)
62 simplrl 773 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑑 ∈ ℕ0)
63 simprl 767 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → ¬ 𝑑 < 𝑐)
64 simprr 769 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))
6511, 59, 60, 61, 62, 63, 64dyadmaxlem 24125 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → (𝑎 = 𝑏𝑐 = 𝑑))
66 oveq12 7154 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 = 𝑏𝑐 = 𝑑) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))
6765, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))
6867exp32 421 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
69 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = (𝑏𝐹𝑑) → ([,]‘𝑤) = ([,]‘(𝑏𝐹𝑑)))
7069sseq2d 3996 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑))))
71 eqeq2 2830 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑏𝐹𝑑) → ((𝑎𝐹𝑐) = 𝑤 ↔ (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))
7270, 71imbi12d 346 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝑏𝐹𝑑) → ((([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
7372imbi2d 342 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑏𝐹𝑑) → ((¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)) ↔ (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))))
7468, 73syl5ibrcom 248 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) → (𝑤 = (𝑏𝐹𝑑) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
7574anassrs 468 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ 𝑑 ∈ ℕ0) ∧ 𝑏 ∈ ℤ) → (𝑤 = (𝑏𝐹𝑑) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
7675rexlimdva 3281 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ 𝑑 ∈ ℕ0) → (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
7776a2d 29 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ 𝑑 ∈ ℕ0) → ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
7877impd 411 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ 𝑑 ∈ ℕ0) → (((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
7978rexlimdva 3281 . . . . . . . . . . . . . . . . 17 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → (∃𝑑 ∈ ℕ0 ((∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ∧ ∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8057, 79syld 47 . . . . . . . . . . . . . . . 16 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → (∀𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8180ralimdva 3174 . . . . . . . . . . . . . . 15 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → (∀𝑤𝐴𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8247, 81syl5bi 243 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → (∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8382imp 407 . . . . . . . . . . . . 13 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))
8483an32s 648 . . . . . . . . . . . 12 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))
85 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑧 = (𝑎𝐹𝑐) → ([,]‘𝑧) = ([,]‘(𝑎𝐹𝑐)))
8685sseq1d 3995 . . . . . . . . . . . . . 14 (𝑧 = (𝑎𝐹𝑐) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤)))
87 eqeq1 2822 . . . . . . . . . . . . . 14 (𝑧 = (𝑎𝐹𝑐) → (𝑧 = 𝑤 ↔ (𝑎𝐹𝑐) = 𝑤))
8886, 87imbi12d 346 . . . . . . . . . . . . 13 (𝑧 = (𝑎𝐹𝑐) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8988ralbidv 3194 . . . . . . . . . . . 12 (𝑧 = (𝑎𝐹𝑐) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
9084, 89syl5ibrcom 248 . . . . . . . . . . 11 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → (𝑧 = (𝑎𝐹𝑐) → ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
9190anassrs 468 . . . . . . . . . 10 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ 𝑧𝐴) ∧ 𝑎 ∈ ℤ) → (𝑧 = (𝑎𝐹𝑐) → ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
9291rexlimdva 3281 . . . . . . . . 9 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ 𝑧𝐴) → (∃𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
9392reximdva 3271 . . . . . . . 8 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
9493ex 413 . . . . . . 7 (((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) → (∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
9543, 94syl5bi 243 . . . . . 6 (((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) → (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
9695com23 86 . . . . 5 (((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
9796expimpd 454 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ((𝑐 ∈ ℕ0 ∧ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)) → (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
9834, 97syl5bi 243 . . 3 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → (𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} → (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))))
9998rexlimdv 3280 . 2 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → (∃𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
10030, 99mpd 15 1 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  ∃!wreu 3137  {crab 3139  Vcvv 3492  cin 3932  wss 3933  c0 4288  cop 4563   class class class wbr 5057   We wwe 5506   × cxp 5546  ran crn 5549   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  cle 10664   / cdiv 11285  2c2 11680  0cn0 11885  cz 11969  cuz 12231  [,]cicc 12729  cexp 13417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-rest 16684  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-top 21430  df-topon 21447  df-bases 21482  df-cmp 21923  df-ovol 23992
This theorem is referenced by:  dyadmbllem  24127
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