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Mirrors > Home > MPE Home > Th. List > bitsp1 | Structured version Visualization version GIF version |
Description: The 𝑀 + 1-th bit of 𝑁 is the 𝑀-th bit of ⌊(𝑁 / 2). (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
bitsp1 | ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘𝑁) ↔ 𝑀 ∈ (bits‘(⌊‘(𝑁 / 2))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 11448 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℕ | |
2 | 1 | a1i 11 | . . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → 2 ∈ ℕ) |
3 | 2 | nncnd 11392 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → 2 ∈ ℂ) |
4 | simpr 479 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → 𝑀 ∈ ℕ0) | |
5 | 3, 4 | expp1d 13328 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (2↑(𝑀 + 1)) = ((2↑𝑀) · 2)) |
6 | 2, 4 | nnexpcld 13351 | . . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (2↑𝑀) ∈ ℕ) |
7 | 6 | nncnd 11392 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (2↑𝑀) ∈ ℂ) |
8 | 7, 3 | mulcomd 10398 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((2↑𝑀) · 2) = (2 · (2↑𝑀))) |
9 | 5, 8 | eqtrd 2814 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (2↑(𝑀 + 1)) = (2 · (2↑𝑀))) |
10 | 9 | oveq2d 6938 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (𝑁 / (2↑(𝑀 + 1))) = (𝑁 / (2 · (2↑𝑀)))) |
11 | simpl 476 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → 𝑁 ∈ ℤ) | |
12 | 11 | zcnd 11835 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → 𝑁 ∈ ℂ) |
13 | 2 | nnne0d 11425 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → 2 ≠ 0) |
14 | 6 | nnne0d 11425 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (2↑𝑀) ≠ 0) |
15 | 12, 3, 7, 13, 14 | divdiv1d 11182 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑁 / 2) / (2↑𝑀)) = (𝑁 / (2 · (2↑𝑀)))) |
16 | 10, 15 | eqtr4d 2817 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (𝑁 / (2↑(𝑀 + 1))) = ((𝑁 / 2) / (2↑𝑀))) |
17 | 16 | fveq2d 6450 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (⌊‘(𝑁 / (2↑(𝑀 + 1)))) = (⌊‘((𝑁 / 2) / (2↑𝑀)))) |
18 | 11 | zred 11834 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → 𝑁 ∈ ℝ) |
19 | 18 | rehalfcld 11629 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (𝑁 / 2) ∈ ℝ) |
20 | fldiv 12978 | . . . . . 6 ⊢ (((𝑁 / 2) ∈ ℝ ∧ (2↑𝑀) ∈ ℕ) → (⌊‘((⌊‘(𝑁 / 2)) / (2↑𝑀))) = (⌊‘((𝑁 / 2) / (2↑𝑀)))) | |
21 | 19, 6, 20 | syl2anc 579 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (⌊‘((⌊‘(𝑁 / 2)) / (2↑𝑀))) = (⌊‘((𝑁 / 2) / (2↑𝑀)))) |
22 | 17, 21 | eqtr4d 2817 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (⌊‘(𝑁 / (2↑(𝑀 + 1)))) = (⌊‘((⌊‘(𝑁 / 2)) / (2↑𝑀)))) |
23 | 22 | breq2d 4898 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (2 ∥ (⌊‘(𝑁 / (2↑(𝑀 + 1)))) ↔ 2 ∥ (⌊‘((⌊‘(𝑁 / 2)) / (2↑𝑀))))) |
24 | 23 | notbid 310 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (¬ 2 ∥ (⌊‘(𝑁 / (2↑(𝑀 + 1)))) ↔ ¬ 2 ∥ (⌊‘((⌊‘(𝑁 / 2)) / (2↑𝑀))))) |
25 | peano2nn0 11684 | . . 3 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0) | |
26 | bitsval2 15553 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 + 1) ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘𝑁) ↔ ¬ 2 ∥ (⌊‘(𝑁 / (2↑(𝑀 + 1)))))) | |
27 | 25, 26 | sylan2 586 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘𝑁) ↔ ¬ 2 ∥ (⌊‘(𝑁 / (2↑(𝑀 + 1)))))) |
28 | 19 | flcld 12918 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (⌊‘(𝑁 / 2)) ∈ ℤ) |
29 | bitsval2 15553 | . . 3 ⊢ (((⌊‘(𝑁 / 2)) ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (𝑀 ∈ (bits‘(⌊‘(𝑁 / 2))) ↔ ¬ 2 ∥ (⌊‘((⌊‘(𝑁 / 2)) / (2↑𝑀))))) | |
30 | 28, 29 | sylancom 582 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (𝑀 ∈ (bits‘(⌊‘(𝑁 / 2))) ↔ ¬ 2 ∥ (⌊‘((⌊‘(𝑁 / 2)) / (2↑𝑀))))) |
31 | 24, 27, 30 | 3bitr4d 303 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘𝑁) ↔ 𝑀 ∈ (bits‘(⌊‘(𝑁 / 2))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 1c1 10273 + caddc 10275 · cmul 10277 / cdiv 11032 ℕcn 11374 2c2 11430 ℕ0cn0 11642 ℤcz 11728 ⌊cfl 12910 ↑cexp 13178 ∥ cdvds 15387 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 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 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 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 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: bitsp1e 15560 bitsp1o 15561 |
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