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Mirrors > Home > MPE Home > Th. List > dyadf | Structured version Visualization version GIF version |
Description: The function 𝐹 returns the endpoints of a dyadic rational covering of the real line. (Contributed by Mario Carneiro, 26-Mar-2015.) |
Ref | Expression |
---|---|
dyadmbl.1 | ⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) |
Ref | Expression |
---|---|
dyadf | ⊢ 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 11986 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
2 | 1 | adantr 483 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → 𝑥 ∈ ℝ) |
3 | 2 | lep1d 11571 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → 𝑥 ≤ (𝑥 + 1)) |
4 | peano2re 10813 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ) | |
5 | 2, 4 | syl 17 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → (𝑥 + 1) ∈ ℝ) |
6 | 2nn 11711 | . . . . . . . . . 10 ⊢ 2 ∈ ℕ | |
7 | nnexpcl 13443 | . . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℕ) | |
8 | 6, 7 | mpan 688 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (2↑𝑦) ∈ ℕ) |
9 | 8 | adantl 484 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℕ) |
10 | 9 | nnred 11653 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℝ) |
11 | 9 | nngt0d 11687 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → 0 < (2↑𝑦)) |
12 | lediv1 11505 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ (𝑥 + 1) ∈ ℝ ∧ ((2↑𝑦) ∈ ℝ ∧ 0 < (2↑𝑦))) → (𝑥 ≤ (𝑥 + 1) ↔ (𝑥 / (2↑𝑦)) ≤ ((𝑥 + 1) / (2↑𝑦)))) | |
13 | 2, 5, 10, 11, 12 | syl112anc 1370 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → (𝑥 ≤ (𝑥 + 1) ↔ (𝑥 / (2↑𝑦)) ≤ ((𝑥 + 1) / (2↑𝑦)))) |
14 | 3, 13 | mpbid 234 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → (𝑥 / (2↑𝑦)) ≤ ((𝑥 + 1) / (2↑𝑦))) |
15 | df-br 5067 | . . . . 5 ⊢ ((𝑥 / (2↑𝑦)) ≤ ((𝑥 + 1) / (2↑𝑦)) ↔ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ ≤ ) | |
16 | 14, 15 | sylib 220 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ ≤ ) |
17 | nndivre 11679 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℕ) → (𝑥 / (2↑𝑦)) ∈ ℝ) | |
18 | 1, 8, 17 | syl2an 597 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → (𝑥 / (2↑𝑦)) ∈ ℝ) |
19 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥 + 1) ∈ ℝ) |
20 | nndivre 11679 | . . . . . 6 ⊢ (((𝑥 + 1) ∈ ℝ ∧ (2↑𝑦) ∈ ℕ) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ) | |
21 | 19, 8, 20 | syl2an 597 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ) |
22 | 18, 21 | opelxpd 5593 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ × ℝ)) |
23 | 16, 22 | elind 4171 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ ( ≤ ∩ (ℝ × ℝ))) |
24 | 23 | rgen2 3203 | . 2 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ ( ≤ ∩ (ℝ × ℝ)) |
25 | dyadmbl.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) | |
26 | 25 | fmpo 7766 | . 2 ⊢ (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))) |
27 | 24, 26 | mpbi 232 | 1 ⊢ 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∩ cin 3935 〈cop 4573 class class class wbr 5066 × cxp 5553 ⟶wf 6351 (class class class)co 7156 ∈ cmpo 7158 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 / cdiv 11297 ℕcn 11638 2c2 11693 ℕ0cn0 11898 ℤcz 11982 ↑cexp 13430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-exp 13431 |
This theorem is referenced by: dyaddisj 24197 dyadmax 24199 dyadmbllem 24200 dyadmbl 24201 opnmbllem 24202 opnmbllem0 34943 mblfinlem2 34945 |
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