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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 15695 | . 2 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ 𝑥 (2↑𝑘) = Σ𝑘 ∈ 𝐴 (2↑𝑘)) | |
2 | bitsinv.k | . . 3 ⊢ 𝐾 = ◡(bits ↾ ℕ0) | |
3 | bitsf1ocnv 16446 | . . . 4 ⊢ ((bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin) ∧ ◡(bits ↾ ℕ0) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 (2↑𝑘))) | |
4 | 3 | simpri 484 | . . 3 ⊢ ◡(bits ↾ ℕ0) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 (2↑𝑘)) |
5 | 2, 4 | eqtri 2754 | . 2 ⊢ 𝐾 = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 (2↑𝑘)) |
6 | sumex 15694 | . 2 ⊢ Σ𝑘 ∈ 𝐴 (2↑𝑘) ∈ V | |
7 | 1, 5, 6 | fvmpt 7011 | 1 ⊢ (𝐴 ∈ (𝒫 ℕ0 ∩ Fin) → (𝐾‘𝐴) = Σ𝑘 ∈ 𝐴 (2↑𝑘)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 𝒫 cpw 4607 ↦ cmpt 5238 ◡ccnv 5683 ↾ cres 5686 –1-1-onto→wf1o 6555 ‘cfv 6556 (class class class)co 7426 Fincfn 8976 2c2 12321 ℕ0cn0 12526 ↑cexp 14083 Σcsu 15692 bitscbits 16421 |
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 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-inf2 9686 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-pre-sup 11238 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-disj 5121 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-se 5640 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-isom 6565 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-2o 8499 df-oadd 8502 df-er 8736 df-map 8859 df-pm 8860 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-sup 9487 df-inf 9488 df-oi 9555 df-dju 9946 df-card 9984 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12599 df-z 12613 df-uz 12877 df-rp 13031 df-fz 13541 df-fzo 13684 df-fl 13814 df-mod 13892 df-seq 14024 df-exp 14084 df-hash 14350 df-cj 15106 df-re 15107 df-im 15108 df-sqrt 15242 df-abs 15243 df-clim 15492 df-sum 15693 df-dvds 16259 df-bits 16424 |
This theorem is referenced by: bitsinvp1 16451 |
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