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Theorem dyadmax 25571
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 13962 . . . . 5 < We (ℤ‘0)
21a1i 11 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → < We (ℤ‘0))
3 nn0ex 12511 . . . . . 6 0 ∈ V
43rabex 5335 . . . . 5 {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V
54a1i 11 . . . 4 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ∈ V)
6 ssrab2 4073 . . . . . 6 {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ⊆ ℕ0
7 nn0uz 12897 . . . . . 6 0 = (ℤ‘0)
86, 7sseqtri 4013 . . . . 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 25564 . . . . . . . . . . 11 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
13 ffn 6723 . . . . . . . . . . 11 (𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ0))
14 ovelrn 7597 . . . . . . . . . . 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 3277 . . . . . . . . . 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 3052 . . . . . . . 8 𝑧 ∈ ran 𝐹𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)
19 ssralv 4045 . . . . . . . 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 4496 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)) → ∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
2210, 20, 21syl2anr 595 . . . . . 6 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃𝑧𝐴𝑛 ∈ ℕ0𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛))
23 rexcom 3277 . . . . . 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 4387 . . . . 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 5674 . . . 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 1369 . . 3 ((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → ∃!𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)}∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐)
29 reurex 3367 . . 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 7427 . . . . . . 7 (𝑛 = 𝑐 → (𝑎𝐹𝑛) = (𝑎𝐹𝑐))
3231eqeq2d 2736 . . . . . 6 (𝑛 = 𝑐 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑧 = (𝑎𝐹𝑐)))
33322rexbidv 3209 . . . . 5 (𝑛 = 𝑐 → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
3433elrab 3679 . . . 4 (𝑐 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ↔ (𝑐 ∈ ℕ0 ∧ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐)))
35 eqeq1 2729 . . . . . . . . . 10 (𝑧 = 𝑤 → (𝑧 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑎𝐹𝑛)))
36 oveq1 7426 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎𝐹𝑛) = (𝑏𝐹𝑛))
3736eqeq2d 2736 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑤 = (𝑎𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑛)))
3835, 37cbvrex2vw 3229 . . . . . . . . 9 (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛) ↔ ∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑛))
39 oveq2 7427 . . . . . . . . . . 11 (𝑛 = 𝑑 → (𝑏𝐹𝑛) = (𝑏𝐹𝑑))
4039eqeq2d 2736 . . . . . . . . . 10 (𝑛 = 𝑑 → (𝑤 = (𝑏𝐹𝑛) ↔ 𝑤 = (𝑏𝐹𝑑)))
41402rexbidv 3209 . . . . . . . . 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 3685 . . . . . . 7 (∀𝑑 ∈ {𝑛 ∈ ℕ0 ∣ ∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑛)} ¬ 𝑑 < 𝑐 ↔ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
44 r19.23v 3172 . . . . . . . . . . . . . . . . 17 (∀𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
4544ralbii 3082 . . . . . . . . . . . . . . . 16 (∀𝑑 ∈ ℕ0𝑤𝐴 (∃𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐) ↔ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐))
46 ralcom 3276 . . . . . . . . . . . . . . . 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 3976 . . . . . . . . . . . . . . . . . . 19 (((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) → 𝑤 ∈ ran 𝐹)
50 ovelrn 7597 . . . . . . . . . . . . . . . . . . . 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 3277 . . . . . . . . . . . . . . . . . . 19 (∃𝑏 ∈ ℤ ∃𝑑 ∈ ℕ0 𝑤 = (𝑏𝐹𝑑) ↔ ∃𝑑 ∈ ℕ0𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑))
54 r19.29 3103 . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . 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 25570 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) ∧ (¬ 𝑑 < 𝑐 ∧ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)))) → (𝑎 = 𝑏𝑐 = 𝑑))
66 oveq12 7428 . . . . . . . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ (𝑑 ∈ ℕ0𝑏 ∈ ℤ)) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
69 fveq2 6896 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = (𝑏𝐹𝑑) → ([,]‘𝑤) = ([,]‘(𝑏𝐹𝑑)))
7069sseq2d 4009 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑏𝐹𝑑) → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑))))
71 eqeq2 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑏𝐹𝑑) → ((𝑎𝐹𝑐) = 𝑤 ↔ (𝑎𝐹𝑐) = (𝑏𝐹𝑑)))
7270, 71imbi12d 343 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = (𝑏𝐹𝑑) → ((([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘(𝑏𝐹𝑑)) → (𝑎𝐹𝑐) = (𝑏𝐹𝑑))))
7372imbi2d 339 . . . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ (𝑧𝐴𝑎 ∈ ℤ)) ∧ 𝑤𝐴) ∧ 𝑑 ∈ ℕ0) ∧ 𝑏 ∈ ℤ) → (𝑤 = (𝑏𝐹𝑑) → (¬ 𝑑 < 𝑐 → (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤))))
7675rexlimdva 3144 . . . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . . 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 3144 . . . . . . . . . . . . . . . . 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 3156 . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . 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 6896 . . . . . . . . . . . . . . 15 (𝑧 = (𝑎𝐹𝑐) → ([,]‘𝑧) = ([,]‘(𝑎𝐹𝑐)))
8685sseq1d 4008 . . . . . . . . . . . . . 14 (𝑧 = (𝑎𝐹𝑐) → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤)))
87 eqeq1 2729 . . . . . . . . . . . . . 14 (𝑧 = (𝑎𝐹𝑐) → (𝑧 = 𝑤 ↔ (𝑎𝐹𝑐) = 𝑤))
8886, 87imbi12d 343 . . . . . . . . . . . . 13 (𝑧 = (𝑎𝐹𝑐) → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘(𝑎𝐹𝑐)) ⊆ ([,]‘𝑤) → (𝑎𝐹𝑐) = 𝑤)))
8988ralbidv 3167 . . . . . . . . . . . 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 466 . . . . . . . . . 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 3144 . . . . . . . . 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 3157 . . . . . . . 8 ((((𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) ∧ 𝑐 ∈ ℕ0) ∧ ∀𝑑 ∈ ℕ0 (∃𝑤𝐴𝑏 ∈ ℤ 𝑤 = (𝑏𝐹𝑑) → ¬ 𝑑 < 𝑐)) → (∃𝑧𝐴𝑎 ∈ ℤ 𝑧 = (𝑎𝐹𝑐) → ∃𝑧𝐴𝑤𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)))
9493ex 411 . . . . . . 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 452 . . . 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 3142 . 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 394   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  ∃!wreu 3361  {crab 3418  Vcvv 3461  cin 3943  wss 3944  c0 4322  cop 4636   class class class wbr 5149   We wwe 5632   × cxp 5676  ran crn 5679   Fn wfn 6544  wf 6545  cfv 6549  (class class class)co 7419  cmpo 7421  cr 11139  0cc0 11140  1c1 11141   + caddc 11143   < clt 11280  cle 11281   / cdiv 11903  2c2 12300  0cn0 12505  cz 12591  cuz 12855  [,]cicc 13362  cexp 14062
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fi 9436  df-sup 9467  df-inf 9468  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-q 12966  df-rp 13010  df-xneg 13127  df-xadd 13128  df-xmul 13129  df-ioo 13363  df-ico 13365  df-icc 13366  df-fz 13520  df-fzo 13663  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-sum 15669  df-rest 17407  df-topgen 17428  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-top 22840  df-topon 22857  df-bases 22893  df-cmp 23335  df-ovol 25437
This theorem is referenced by:  dyadmbllem  25572
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