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Mirrors > Home > MPE Home > Th. List > bitsp1e | Structured version Visualization version GIF version |
Description: The 𝑀 + 1-th bit of 2𝑁 is the 𝑀-th bit of 𝑁. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
bitsp1e | ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘(2 · 𝑁)) ↔ 𝑀 ∈ (bits‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 12618 | . . . . 5 ⊢ 2 ∈ ℤ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℤ) |
3 | id 22 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℤ) | |
4 | 2, 3 | zmulcld 12696 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) ∈ ℤ) |
5 | bitsp1 16399 | . . 3 ⊢ (((2 · 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘(2 · 𝑁)) ↔ 𝑀 ∈ (bits‘(⌊‘((2 · 𝑁) / 2))))) | |
6 | 4, 5 | sylan 579 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘(2 · 𝑁)) ↔ 𝑀 ∈ (bits‘(⌊‘((2 · 𝑁) / 2))))) |
7 | zcn 12587 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
8 | 2cnd 12314 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
9 | 2ne0 12340 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
10 | 9 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 2 ≠ 0) |
11 | 7, 8, 10 | divcan3d 12019 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → ((2 · 𝑁) / 2) = 𝑁) |
12 | 11 | fveq2d 6895 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘((2 · 𝑁) / 2)) = (⌊‘𝑁)) |
13 | flid 13799 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (⌊‘𝑁) = 𝑁) | |
14 | 12, 13 | eqtrd 2768 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (⌊‘((2 · 𝑁) / 2)) = 𝑁) |
15 | 14 | adantr 480 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (⌊‘((2 · 𝑁) / 2)) = 𝑁) |
16 | 15 | fveq2d 6895 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (bits‘(⌊‘((2 · 𝑁) / 2))) = (bits‘𝑁)) |
17 | 16 | eleq2d 2815 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (𝑀 ∈ (bits‘(⌊‘((2 · 𝑁) / 2))) ↔ 𝑀 ∈ (bits‘𝑁))) |
18 | 6, 17 | bitrd 279 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → ((𝑀 + 1) ∈ (bits‘(2 · 𝑁)) ↔ 𝑀 ∈ (bits‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ‘cfv 6542 (class class class)co 7414 0cc0 11132 1c1 11133 + caddc 11135 · cmul 11137 / cdiv 11895 2c2 12291 ℕ0cn0 12496 ℤcz 12582 ⌊cfl 13781 bitscbits 16387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-n0 12497 df-z 12583 df-uz 12847 df-fl 13783 df-seq 13993 df-exp 14053 df-bits 16390 |
This theorem is referenced by: (None) |
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