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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 12015 | . . . . . 6 ⊢ 2 ∈ ℤ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℤ) |
3 | id 22 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℤ) | |
4 | 2, 3 | zmulcld 12094 | . . . 4 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℤ) |
5 | 4 | peano2zd 12091 | . . 3 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) + 1) ∈ ℤ) |
6 | bitsp1 15780 | . . 3 ⊢ ((((2 · 𝑁) + 1) ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘((2 · 𝑁) + 1)) ↔ 𝑀 ∈ (bits‘(⌊‘(((2 · 𝑁) + 1) / 2))))) | |
7 | 5, 6 | sylan 582 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘((2 · 𝑁) + 1)) ↔ 𝑀 ∈ (bits‘(⌊‘(((2 · 𝑁) + 1) / 2))))) |
8 | 2re 11712 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
9 | 8 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
10 | zre 11986 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
11 | 9, 10 | remulcld 10671 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℝ) |
12 | 11 | recnd 10669 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℂ) |
13 | 1cnd 10636 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
14 | 2cnd 11716 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
15 | 2ne0 11742 | . . . . . . . . . 10 ⊢ 2 ≠ 0 | |
16 | 15 | a1i 11 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 2 ≠ 0) |
17 | 12, 13, 14, 16 | divdird 11454 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) + 1) / 2) = (((2 · 𝑁) / 2) + (1 / 2))) |
18 | zcn 11987 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
19 | 18, 14, 16 | divcan3d 11421 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) / 2) = 𝑁) |
20 | 19 | oveq1d 7171 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) / 2) + (1 / 2)) = (𝑁 + (1 / 2))) |
21 | 17, 20 | eqtrd 2856 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → (((2 · 𝑁) + 1) / 2) = (𝑁 + (1 / 2))) |
22 | 21 | fveq2d 6674 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘(((2 · 𝑁) + 1) / 2)) = (⌊‘(𝑁 + (1 / 2)))) |
23 | halfge0 11855 | . . . . . . . 8 ⊢ 0 ≤ (1 / 2) | |
24 | halflt1 11856 | . . . . . . . 8 ⊢ (1 / 2) < 1 | |
25 | 23, 24 | pm3.2i 473 | . . . . . . 7 ⊢ (0 ≤ (1 / 2) ∧ (1 / 2) < 1) |
26 | halfre 11852 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ | |
27 | flbi2 13188 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ (1 / 2) ∈ ℝ) → ((⌊‘(𝑁 + (1 / 2))) = 𝑁 ↔ (0 ≤ (1 / 2) ∧ (1 / 2) < 1))) | |
28 | 26, 27 | mpan2 689 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → ((⌊‘(𝑁 + (1 / 2))) = 𝑁 ↔ (0 ≤ (1 / 2) ∧ (1 / 2) < 1))) |
29 | 25, 28 | mpbiri 260 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 + (1 / 2))) = 𝑁) |
30 | 22, 29 | eqtrd 2856 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (⌊‘(((2 · 𝑁) + 1) / 2)) = 𝑁) |
31 | 30 | adantr 483 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (⌊‘(((2 · 𝑁) + 1) / 2)) = 𝑁) |
32 | 31 | fveq2d 6674 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (bits‘(⌊‘(((2 · 𝑁) + 1) / 2))) = (bits‘𝑁)) |
33 | 32 | eleq2d 2898 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (𝑀 ∈ (bits‘(⌊‘(((2 · 𝑁) + 1) / 2))) ↔ 𝑀 ∈ (bits‘𝑁))) |
34 | 7, 33 | bitrd 281 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 < clt 10675 ≤ cle 10676 / cdiv 11297 2c2 11693 ℕ0cn0 11898 ℤcz 11982 ⌊cfl 13161 bitscbits 15768 |
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 ax-pre-sup 10615 |
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-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 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-fl 13163 df-seq 13371 df-exp 13431 df-bits 15771 |
This theorem is referenced by: (None) |
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