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Mirrors > Home > MPE Home > Th. List > bitsp1o | Structured version Visualization version GIF version |
Description: The 𝑀 + 1-th bit of 2𝑁 + 1 is the 𝑀-th bit of 𝑁. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
bitsp1o | ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘((2 · 𝑁) + 1)) ↔ 𝑀 ∈ (bits‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 11761 | . . . . . 6 ⊢ 2 ∈ ℤ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℤ) |
3 | id 22 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℤ) | |
4 | 2, 3 | zmulcld 11840 | . . . 4 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℤ) |
5 | 4 | peano2zd 11837 | . . 3 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) + 1) ∈ ℤ) |
6 | bitsp1 15559 | . . 3 ⊢ ((((2 · 𝑁) + 1) ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘((2 · 𝑁) + 1)) ↔ 𝑀 ∈ (bits‘(⌊‘(((2 · 𝑁) + 1) / 2))))) | |
7 | 5, 6 | sylan 575 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘((2 · 𝑁) + 1)) ↔ 𝑀 ∈ (bits‘(⌊‘(((2 · 𝑁) + 1) / 2))))) |
8 | 2re 11449 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
9 | 8 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
10 | zre 11732 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
11 | 9, 10 | remulcld 10407 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℝ) |
12 | 11 | recnd 10405 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℂ) |
13 | 1cnd 10371 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
14 | 2cnd 11453 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
15 | 2ne0 11486 | . . . . . . . . . 10 ⊢ 2 ≠ 0 | |
16 | 15 | a1i 11 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 2 ≠ 0) |
17 | 12, 13, 14, 16 | divdird 11189 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) + 1) / 2) = (((2 · 𝑁) / 2) + (1 / 2))) |
18 | zcn 11733 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
19 | 18, 14, 16 | divcan3d 11156 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) / 2) = 𝑁) |
20 | 19 | oveq1d 6937 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) / 2) + (1 / 2)) = (𝑁 + (1 / 2))) |
21 | 17, 20 | eqtrd 2813 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) + 1) / 2) = (𝑁 + (1 / 2))) |
22 | 21 | fveq2d 6450 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘(((2 · 𝑁) + 1) / 2)) = (⌊‘(𝑁 + (1 / 2)))) |
23 | halfge0 11599 | . . . . . . . 8 ⊢ 0 ≤ (1 / 2) | |
24 | halflt1 11600 | . . . . . . . 8 ⊢ (1 / 2) < 1 | |
25 | 23, 24 | pm3.2i 464 | . . . . . . 7 ⊢ (0 ≤ (1 / 2) ∧ (1 / 2) < 1) |
26 | halfre 11596 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ | |
27 | flbi2 12937 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ (1 / 2) ∈ ℝ) → ((⌊‘(𝑁 + (1 / 2))) = 𝑁 ↔ (0 ≤ (1 / 2) ∧ (1 / 2) < 1))) | |
28 | 26, 27 | mpan2 681 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → ((⌊‘(𝑁 + (1 / 2))) = 𝑁 ↔ (0 ≤ (1 / 2) ∧ (1 / 2) < 1))) |
29 | 25, 28 | mpbiri 250 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 + (1 / 2))) = 𝑁) |
30 | 22, 29 | eqtrd 2813 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (⌊‘(((2 · 𝑁) + 1) / 2)) = 𝑁) |
31 | 30 | adantr 474 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (⌊‘(((2 · 𝑁) + 1) / 2)) = 𝑁) |
32 | 31 | fveq2d 6450 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (bits‘(⌊‘(((2 · 𝑁) + 1) / 2))) = (bits‘𝑁)) |
33 | 32 | eleq2d 2844 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (𝑀 ∈ (bits‘(⌊‘(((2 · 𝑁) + 1) / 2))) ↔ 𝑀 ∈ (bits‘𝑁))) |
34 | 7, 33 | bitrd 271 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘((2 · 𝑁) + 1)) ↔ 𝑀 ∈ (bits‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 0cc0 10272 1c1 10273 + caddc 10275 · cmul 10277 < clt 10411 ≤ cle 10412 / cdiv 11032 2c2 11430 ℕ0cn0 11642 ℤcz 11728 ⌊cfl 12910 bitscbits 15547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-n0 11643 df-z 11729 df-uz 11993 df-fl 12912 df-seq 13120 df-exp 13179 df-bits 15550 |
This theorem is referenced by: (None) |
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