MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dyadmax Structured version   Visualization version   GIF version

Theorem dyadmax 23659
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 12971 . . . . 5 < We (ℤ‘0)
21a1i 11 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → < We (ℤ‘0))
3 nn0ex 11547 . . . . . 6 0 ∈ V
43rabex 4975 . . . . 5 {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V
54a1i 11 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V)
6 ssrab2 3849 . . . . . 6 {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ ℕ0
7 nn0uz 11925 . . . . . 6 0 = (ℤ‘0)
86, 7sseqtri 3799 . . . . 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 23652 . . . . . . . . . . 11 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
13 ffn 6225 . . . . . . . . . . 11 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14 ovelrn 7010 . . . . . . . . . . 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 3246 . . . . . . . . . 10 (∃𝑎 ∈ ℤ ∃𝑛 ∈ ℕ0 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
1715, 16sylbb 210 . . . . . . . . 9 (𝑧 ∈ ran 𝐹 → ∃𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
1817rgen 3069 . . . . . . . 8 𝑧 ∈ ran 𝐹𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)
19 ssralv 3828 . . . . . . . 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 4221 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)) → ∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2210, 20, 21syl2anr 590 . . . . . 6 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
23 rexcom 3246 . . . . . 6 (∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑛 ∈ ℕ0𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2422, 23sylib 209 . . . . 5 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑛 ∈ ℕ0𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
25 rabn0 4124 . . . . 5 ({𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅ ↔ ∃𝑛 ∈ ℕ0𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2624, 25sylibr 225 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ≠ ∅)
27 wereu 5275 . . . 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 1491 . . 3 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃!𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
29 reurex 3308 . . 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 6852 . . . . . . 7 (𝑛 = 𝑐 → (𝑎𝐹𝑛) = (𝑎𝐹𝑐))
3231eqeq2d 2775 . . . . . 6 (𝑛 = 𝑐 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑧 = (𝑎𝐹𝑐)))
33322rexbidv 3204 . . . . 5 (𝑛 = 𝑐 → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
3433elrab 3521 . . . 4 (𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ↔ (𝑐 ∈ ℕ0 ∧ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
35 eqeq1 2769 . . . . . . . . . 10 (𝑧 = 𝑤 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑎𝐹𝑛)))
36 oveq1 6851 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎𝐹𝑛) = (𝑏𝐹𝑛))
3736eqeq2d 2775 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑤 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑛)))
3835, 37cbvrex2v 3328 . . . . . . . . 9 (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛))
39 oveq2 6852 . . . . . . . . . . 11 (𝑛 = 𝑑 → (𝑏𝐹𝑛) = (𝑏𝐹𝑑))
4039eqeq2d 2775 . . . . . . . . . 10 (𝑛 = 𝑑 → (𝑤 = (𝑏𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑑)))
41402rexbidv 3204 . . . . . . . . 9 (𝑛 = 𝑑 → (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
4238, 41syl5bb 274 . . . . . . . 8 (𝑛 = 𝑑 → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑)))
4342ralrab 3527 . . . . . . 7 (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 ↔ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
44 r19.23v 3170 . . . . . . . . . . . . . . . . 17 (∀𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
4544ralbii 3127 . . . . . . . . . . . . . . . 16 (∀𝑑 ∈ ℕ0𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
46 ralcom 3245 . . . . . . . . . . . . . . . 16 (∀𝑑 ∈ ℕ0𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤𝐴𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
4745, 46bitr3i 268 . . . . . . . . . . . . . . 15 (∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑤𝐴𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
48 simplll 791 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → 𝐴 ⊆ ran 𝐹)
4948sselda 3763 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → 𝑤 ∈ ran 𝐹)
50 ovelrn 7010 . . . . . . . . . . . . . . . . . . . 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 209 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → ∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑))
53 rexcom 3246 . . . . . . . . . . . . . . . . . . 19 (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) ↔ ∃𝑑 ∈ ℕ0𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))
54 r19.29 3219 . . . . . . . . . . . . . . . . . . . 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 402 . . . . . . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . . . . . . 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 796 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → 𝑎 ∈ ℤ)
5958ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑎 ∈ ℤ)
60 simplrr 796 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑏 ∈ ℤ)
61 simp-5r 807 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑐 ∈ ℕ0)
62 simplrl 795 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → 𝑑 ∈ ℕ0)
63 simprl 787 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → ¬ 𝑑 < 𝑐)
64 simprr 789 . . . . . . . . . . . . . . . . . . . . . . . . . 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 23658 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → (𝑎 = 𝑏𝑐 = 𝑑))
66 oveq12 6853 . . . . . . . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
69 fveq2 6377 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = (𝑏𝐹𝑑) → ([,]‘𝑤) = ([,]‘(𝑏𝐹𝑑)))
7069sseq2d 3795 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑))))
71 eqeq2 2776 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑏𝐹𝑑) → ((𝑎𝐹𝑐) = 𝑤 ↔ (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))
7270, 71imbi12d 335 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝑏𝐹𝑑) → ((([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
7372imbi2d 331 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = (𝑏𝐹𝑑) → ((¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)) ↔ (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))))
7468, 73syl5ibrcom 238 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) → (𝑤 = (𝑏𝐹𝑑) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
7574anassrs 459 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ 𝑑 ∈ ℕ0) ∧ 𝑏 ∈ ℤ) → (𝑤 = (𝑏𝐹𝑑) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
7675rexlimdva 3178 . . . . . . . . . . . . . . . . . . . 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 398 . . . . . . . . . . . . . . . . . 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 3178 . . . . . . . . . . . . . . . . 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 3109 . . . . . . . . . . . . . . 15 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → (∀𝑤𝐴𝑑 ∈ ℕ0 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8247, 81syl5bi 233 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → (∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8382imp 395 . . . . . . . . . . . . 13 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))
8483an32s 642 . . . . . . . . . . . 12 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))
85 fveq2 6377 . . . . . . . . . . . . . . 15 (𝑧 = (𝑎𝐹𝑐) → ([,]‘𝑧) = ([,]‘(𝑎𝐹𝑐)))
8685sseq1d 3794 . . . . . . . . . . . . . 14 (𝑧 = (𝑎𝐹𝑐) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤)))
87 eqeq1 2769 . . . . . . . . . . . . . 14 (𝑧 = (𝑎𝐹𝑐) → (𝑧 = 𝑤 ↔ (𝑎𝐹𝑐) = 𝑤))
8886, 87imbi12d 335 . . . . . . . . . . . . 13 (𝑧 = (𝑎𝐹𝑐) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8988ralbidv 3133 . . . . . . . . . . . 12 (𝑧 = (𝑎𝐹𝑐) → (∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐴 (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
9084, 89syl5ibrcom 238 . . . . . . . . . . 11 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) ∧ (𝑧𝐴𝑎 ∈ ℤ)) → (𝑧 = (𝑎𝐹𝑐) → ∀𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
9190anassrs 459 . . . . . . . . . 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 3178 . . . . . . . . 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 3163 . . . . . . . 8 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
9493ex 401 . . . . . . 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 233 . . . . . 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 445 . . . 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 233 . . 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 3177 . 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 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  ∃!wreu 3057  {crab 3059  Vcvv 3350  cin 3733  wss 3734  c0 4081  cop 4342   class class class wbr 4811   We wwe 5237   × cxp 5277  ran crn 5280   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6844  cmpt2 6846  cr 10190  0cc0 10191  1c1 10192   + caddc 10194   < clt 10330  cle 10331   / cdiv 10940  2c2 11329  0cn0 11540  cz 11626  cuz 11889  [,]cicc 12383  cexp 13070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-inf2 8755  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-pre-sup 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-oadd 7770  df-er 7949  df-map 8064  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-fi 8526  df-sup 8557  df-inf 8558  df-oi 8624  df-card 9018  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-n0 11541  df-z 11627  df-uz 11890  df-q 11993  df-rp 12032  df-xneg 12149  df-xadd 12150  df-xmul 12151  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12537  df-fzo 12677  df-seq 13012  df-exp 13071  df-hash 13325  df-cj 14127  df-re 14128  df-im 14129  df-sqrt 14263  df-abs 14264  df-clim 14507  df-sum 14705  df-rest 16352  df-topgen 16373  df-psmet 20014  df-xmet 20015  df-met 20016  df-bl 20017  df-mopn 20018  df-top 20981  df-topon 20998  df-bases 21033  df-cmp 21473  df-ovol 23525
This theorem is referenced by:  dyadmbllem  23660
  Copyright terms: Public domain W3C validator