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Mirrors > Home > MPE Home > Th. List > bitsinv | Structured version Visualization version GIF version |
Description: The inverse of the bits function. (Contributed by Mario Carneiro, 8-Sep-2016.) |
Ref | Expression |
---|---|
bitsinv.k | ⊢ 𝐾 = ◡(bits ↾ ℕ0) |
Ref | Expression |
---|---|
bitsinv | ⊢ (𝐴 ∈ (𝒫 ℕ0 ∩ Fin) → (𝐾‘𝐴) = Σ𝑘 ∈ 𝐴 (2↑𝑘)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumeq1 14760 | . 2 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ 𝑥 (2↑𝑘) = Σ𝑘 ∈ 𝐴 (2↑𝑘)) | |
2 | bitsinv.k | . . 3 ⊢ 𝐾 = ◡(bits ↾ ℕ0) | |
3 | bitsf1ocnv 15501 | . . . 4 ⊢ ((bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin) ∧ ◡(bits ↾ ℕ0) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 (2↑𝑘))) | |
4 | 3 | simpri 480 | . . 3 ⊢ ◡(bits ↾ ℕ0) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 (2↑𝑘)) |
5 | 2, 4 | eqtri 2821 | . 2 ⊢ 𝐾 = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 (2↑𝑘)) |
6 | sumex 14759 | . 2 ⊢ Σ𝑘 ∈ 𝐴 (2↑𝑘) ∈ V | |
7 | 1, 5, 6 | fvmpt 6507 | 1 ⊢ (𝐴 ∈ (𝒫 ℕ0 ∩ Fin) → (𝐾‘𝐴) = Σ𝑘 ∈ 𝐴 (2↑𝑘)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ∩ cin 3768 𝒫 cpw 4349 ↦ cmpt 4922 ◡ccnv 5311 ↾ cres 5314 –1-1-onto→wf1o 6100 ‘cfv 6101 (class class class)co 6878 Fincfn 8195 2c2 11368 ℕ0cn0 11580 ↑cexp 13114 Σcsu 14757 bitscbits 15476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-disj 4812 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-xnn0 11653 df-z 11667 df-uz 11931 df-rp 12075 df-fz 12581 df-fzo 12721 df-fl 12848 df-mod 12924 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-clim 14560 df-sum 14758 df-dvds 15320 df-bits 15479 |
This theorem is referenced by: bitsinvp1 15506 |
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