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Theorem dyadmax 24962
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 13866 . . . . 5 < We (ℤ‘0)
21a1i 11 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → < We (ℤ‘0))
3 nn0ex 12419 . . . . . 6 0 ∈ V
43rabex 5289 . . . . 5 {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V
54a1i 11 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V)
6 ssrab2 4037 . . . . . 6 {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ ℕ0
7 nn0uz 12805 . . . . . 6 0 = (ℤ‘0)
86, 7sseqtri 3980 . . . . 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 24955 . . . . . . . . . . 11 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
13 ffn 6668 . . . . . . . . . . 11 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14 ovelrn 7530 . . . . . . . . . . 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 3273 . . . . . . . . . 10 (∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
1715, 16sylbb 218 . . . . . . . . 9 (𝑧 ∈ ran 𝐹 → ∃𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
1817rgen 3066 . . . . . . . 8 𝑧 ∈ ran 𝐹𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)
19 ssralv 4010 . . . . . . . 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 4452 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)) → ∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2210, 20, 21syl2anr 597 . . . . . 6 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
23 rexcom 3273 . . . . . 6 (∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2422, 23sylib 217 . . . . 5 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑛 ∈ ℕ0𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
25 rabn0 4345 . . . . 5 ({𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅ ↔ ∃𝑛 ∈ ℕ0𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2624, 25sylibr 233 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅)
27 wereu 5629 . . . 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 1372 . . 3 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃!𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
29 reurex 3357 . . 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 7365 . . . . . . 7 (𝑛 = 𝑐 → (𝑎𝐹𝑛) = (𝑎𝐹𝑐))
3231eqeq2d 2747 . . . . . 6 (𝑛 = 𝑐 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑧 = (𝑎𝐹𝑐)))
33322rexbidv 3213 . . . . 5 (𝑛 = 𝑐 → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
3433elrab 3645 . . . 4 (𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ↔ (𝑐 ∈ ℕ0 ∧ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
35 eqeq1 2740 . . . . . . . . . 10 (𝑧 = 𝑤 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑎𝐹𝑛)))
36 oveq1 7364 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎𝐹𝑛) = (𝑏𝐹𝑛))
3736eqeq2d 2747 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑤 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑛)))
3835, 37cbvrex2vw 3228 . . . . . . . . 9 (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛))
39 oveq2 7365 . . . . . . . . . . 11 (𝑛 = 𝑑 → (𝑏𝐹𝑛) = (𝑏𝐹𝑑))
4039eqeq2d 2747 . . . . . . . . . 10 (𝑛 = 𝑑 → (𝑤 = (𝑏𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑑)))
41402rexbidv 3213 . . . . . . . . 9 (𝑛 = 𝑑 → (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
4238, 41bitrid 282 . . . . . . . 8 (𝑛 = 𝑑 → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
4342ralrab 3651 . . . . . . 7 (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 ↔ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
44 r19.23v 3179 . . . . . . . . . . . . . . . . 17 (∀𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
4544ralbii 3096 . . . . . . . . . . . . . . . 16 (∀𝑑 ∈ ℕ0𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
46 ralcom 3272 . . . . . . . . . . . . . . . 16 (∀𝑑 ∈ ℕ0𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤𝐴𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
4745, 46bitr3i 276 . . . . . . . . . . . . . . 15 (∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤𝐴𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
48 simplll 773 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → 𝐴 ⊆ ran 𝐹)
4948sselda 3944 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → 𝑤 ∈ ran 𝐹)
50 ovelrn 7530 . . . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑))
53 rexcom 3273 . . . . . . . . . . . . . . . . . . 19 (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) ↔ ∃𝑑 ∈ ℕ0𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))
54 r19.29 3117 . . . . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → 𝑎 ∈ ℤ)
5958ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑎 ∈ ℤ)
60 simplrr 776 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑏 ∈ ℤ)
61 simp-5r 784 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑐 ∈ ℕ0)
62 simplrl 775 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑑 ∈ ℕ0)
63 simprl 769 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → ¬ 𝑑 < 𝑐)
64 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . 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 24961 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → (𝑎 = 𝑏𝑐 = 𝑑))
66 oveq12 7366 . . . . . . . . . . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = (𝑏𝐹𝑑) → ([,]‘𝑤) = ([,]‘(𝑏𝐹𝑑)))
7069sseq2d 3976 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑))))
71 eqeq2 2748 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑏𝐹𝑑) → ((𝑎𝐹𝑐) = 𝑤 ↔ (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))
7270, 71imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝑏𝐹𝑑) → ((([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
7372imbi2d 340 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑏𝐹𝑑) → ((¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)) ↔ (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))))
7468, 73syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . . . 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 3152 . . . . . . . . . . . . . . . . . . . 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 3152 . . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . 15 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → (∀𝑤𝐴𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8247, 81biimtrid 241 . . . . . . . . . . . . . 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 650 . . . . . . . . . . . 12 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))
85 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑧 = (𝑎𝐹𝑐) → ([,]‘𝑧) = ([,]‘(𝑎𝐹𝑐)))
8685sseq1d 3975 . . . . . . . . . . . . . 14 (𝑧 = (𝑎𝐹𝑐) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤)))
87 eqeq1 2740 . . . . . . . . . . . . . 14 (𝑧 = (𝑎𝐹𝑐) → (𝑧 = 𝑤 ↔ (𝑎𝐹𝑐) = 𝑤))
8886, 87imbi12d 344 . . . . . . . . . . . . 13 (𝑧 = (𝑎𝐹𝑐) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8988ralbidv 3174 . . . . . . . . . . . 12 (𝑧 = (𝑎𝐹𝑐) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
9084, 89syl5ibrcom 246 . . . . . . . . . . 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 3152 . . . . . . . . 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 3165 . . . . . . . 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, 94biimtrid 241 . . . . . 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, 97biimtrid 241 . . 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 3150 . 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 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  ∃!wreu 3351  {crab 3407  Vcvv 3445  cin 3909  wss 3910  c0 4282  cop 4592   class class class wbr 5105   We wwe 5587   × cxp 5631  ran crn 5634   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cle 11190   / cdiv 11812  2c2 12208  0cn0 12413  cz 12499  cuz 12763  [,]cicc 13267  cexp 13967
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-inf2 9577  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  ax-pre-sup 11129
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-rmo 3353  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-se 5589  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-isom 6505  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-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  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-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828
This theorem is referenced by:  dyadmbllem  24963
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