![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 2ebits | Structured version Visualization version GIF version |
Description: The bits of a power of two. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
2ebits | ⊢ (𝑁 ∈ ℕ0 → (bits‘(2↑𝑁)) = {𝑁}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 12332 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℕ) |
3 | id 22 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
4 | 2, 3 | nnexpcld 14257 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℕ) |
5 | 4 | nncnd 12275 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℂ) |
6 | oveq2 7431 | . . . . 5 ⊢ (𝑘 = 𝑁 → (2↑𝑘) = (2↑𝑁)) | |
7 | 6 | sumsn 15745 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (2↑𝑁) ∈ ℂ) → Σ𝑘 ∈ {𝑁} (2↑𝑘) = (2↑𝑁)) |
8 | 5, 7 | mpdan 685 | . . 3 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ {𝑁} (2↑𝑘) = (2↑𝑁)) |
9 | 8 | fveq2d 6904 | . 2 ⊢ (𝑁 ∈ ℕ0 → (bits‘Σ𝑘 ∈ {𝑁} (2↑𝑘)) = (bits‘(2↑𝑁))) |
10 | snssi 4816 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑁} ⊆ ℕ0) | |
11 | snfi 9080 | . . . 4 ⊢ {𝑁} ∈ Fin | |
12 | elfpw 9394 | . . . 4 ⊢ ({𝑁} ∈ (𝒫 ℕ0 ∩ Fin) ↔ ({𝑁} ⊆ ℕ0 ∧ {𝑁} ∈ Fin)) | |
13 | 10, 11, 12 | sylanblrc 588 | . . 3 ⊢ (𝑁 ∈ ℕ0 → {𝑁} ∈ (𝒫 ℕ0 ∩ Fin)) |
14 | bitsinv2 16438 | . . 3 ⊢ ({𝑁} ∈ (𝒫 ℕ0 ∩ Fin) → (bits‘Σ𝑘 ∈ {𝑁} (2↑𝑘)) = {𝑁}) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → (bits‘Σ𝑘 ∈ {𝑁} (2↑𝑘)) = {𝑁}) |
16 | 9, 15 | eqtr3d 2767 | 1 ⊢ (𝑁 ∈ ℕ0 → (bits‘(2↑𝑁)) = {𝑁}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3945 ⊆ wss 3946 𝒫 cpw 4606 {csn 4632 ‘cfv 6553 (class class class)co 7423 Fincfn 8973 ℂcc 11152 ℕcn 12259 2c2 12314 ℕ0cn0 12519 ↑cexp 14076 Σcsu 15685 bitscbits 16414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-inf2 9680 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-disj 5118 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-oadd 8499 df-er 8733 df-map 8856 df-pm 8857 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9481 df-inf 9482 df-oi 9549 df-dju 9940 df-card 9978 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-n0 12520 df-xnn0 12592 df-z 12606 df-uz 12870 df-rp 13024 df-fz 13534 df-fzo 13677 df-fl 13807 df-mod 13885 df-seq 14017 df-exp 14077 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-sum 15686 df-dvds 16252 df-bits 16417 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |