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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 12499 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
2 | 1 | adantr 481 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → 𝑥 ∈ ℝ) |
3 | 2 | lep1d 12082 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → 𝑥 ≤ (𝑥 + 1)) |
4 | peano2re 11324 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ) | |
5 | 2, 4 | syl 17 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → (𝑥 + 1) ∈ ℝ) |
6 | 2nn 12222 | . . . . . . . . . 10 ⊢ 2 ∈ ℕ | |
7 | nnexpcl 13972 | . . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℕ) | |
8 | 6, 7 | mpan 688 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0 → (2↑𝑦) ∈ ℕ) |
9 | 8 | adantl 482 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℕ) |
10 | 9 | nnred 12164 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℝ) |
11 | 9 | nngt0d 12198 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → 0 < (2↑𝑦)) |
12 | lediv1 12016 | . . . . . . 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 1374 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → (𝑥 ≤ (𝑥 + 1) ↔ (𝑥 / (2↑𝑦)) ≤ ((𝑥 + 1) / (2↑𝑦)))) |
14 | 3, 13 | mpbid 231 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → (𝑥 / (2↑𝑦)) ≤ ((𝑥 + 1) / (2↑𝑦))) |
15 | df-br 5104 | . . . . 5 ⊢ ((𝑥 / (2↑𝑦)) ≤ ((𝑥 + 1) / (2↑𝑦)) ↔ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ ≤ ) | |
16 | 14, 15 | sylib 217 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ ≤ ) |
17 | nndivre 12190 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℕ) → (𝑥 / (2↑𝑦)) ∈ ℝ) | |
18 | 1, 8, 17 | syl2an 596 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → (𝑥 / (2↑𝑦)) ∈ ℝ) |
19 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥 + 1) ∈ ℝ) |
20 | nndivre 12190 | . . . . . 6 ⊢ (((𝑥 + 1) ∈ ℝ ∧ (2↑𝑦) ∈ ℕ) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ) | |
21 | 19, 8, 20 | syl2an 596 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ) |
22 | 18, 21 | opelxpd 5669 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ × ℝ)) |
23 | 16, 22 | elind 4152 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ ( ≤ ∩ (ℝ × ℝ))) |
24 | 23 | rgen2 3192 | . 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 7996 | . 2 ⊢ (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))) |
27 | 24, 26 | mpbi 229 | 1 ⊢ 𝐹:(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ∩ cin 3907 〈cop 4590 class class class wbr 5103 × cxp 5629 ⟶wf 6489 (class class class)co 7353 ∈ cmpo 7355 ℝcr 11046 0cc0 11047 1c1 11048 + caddc 11050 < clt 11185 ≤ cle 11186 / cdiv 11808 ℕcn 12149 2c2 12204 ℕ0cn0 12409 ℤcz 12495 ↑cexp 13959 |
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-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
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 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-n0 12410 df-z 12496 df-uz 12760 df-seq 13899 df-exp 13960 |
This theorem is referenced by: dyaddisj 24944 dyadmax 24946 dyadmbllem 24947 dyadmbl 24948 opnmbllem 24949 opnmbllem0 36081 mblfinlem2 36083 |
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