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| Description: Every number is associated with a finite set of bits. (Contributed by Mario Carneiro, 5-Sep-2016.) | 
| Ref | Expression | 
|---|---|
| bitsfi | ⊢ (𝑁 ∈ ℕ0 → (bits‘𝑁) ∈ Fin) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nn0re 12537 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
| 2 | 2re 12341 | . . . 4 ⊢ 2 ∈ ℝ | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) | 
| 4 | 1lt2 12438 | . . . 4 ⊢ 1 < 2 | |
| 5 | 4 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 1 < 2) | 
| 6 | expnbnd 14272 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ 2 ∈ ℝ ∧ 1 < 2) → ∃𝑚 ∈ ℕ 𝑁 < (2↑𝑚)) | |
| 7 | 1, 3, 5, 6 | syl3anc 1372 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∃𝑚 ∈ ℕ 𝑁 < (2↑𝑚)) | 
| 8 | fzofi 14016 | . . 3 ⊢ (0..^𝑚) ∈ Fin | |
| 9 | simpl 482 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 ∈ ℕ0) | |
| 10 | nn0uz 12921 | . . . . . 6 ⊢ ℕ0 = (ℤ≥‘0) | |
| 11 | 9, 10 | eleqtrdi 2850 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 ∈ (ℤ≥‘0)) | 
| 12 | 2nn 12340 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
| 13 | 12 | a1i 11 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 2 ∈ ℕ) | 
| 14 | simprl 770 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑚 ∈ ℕ) | |
| 15 | 14 | nnnn0d 12589 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑚 ∈ ℕ0) | 
| 16 | 13, 15 | nnexpcld 14285 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → (2↑𝑚) ∈ ℕ) | 
| 17 | 16 | nnzd 12642 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → (2↑𝑚) ∈ ℤ) | 
| 18 | simprr 772 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 < (2↑𝑚)) | |
| 19 | elfzo2 13703 | . . . . 5 ⊢ (𝑁 ∈ (0..^(2↑𝑚)) ↔ (𝑁 ∈ (ℤ≥‘0) ∧ (2↑𝑚) ∈ ℤ ∧ 𝑁 < (2↑𝑚))) | |
| 20 | 11, 17, 18, 19 | syl3anbrc 1343 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 ∈ (0..^(2↑𝑚))) | 
| 21 | 9 | nn0zd 12641 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 ∈ ℤ) | 
| 22 | bitsfzo 16473 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0) → (𝑁 ∈ (0..^(2↑𝑚)) ↔ (bits‘𝑁) ⊆ (0..^𝑚))) | |
| 23 | 21, 15, 22 | syl2anc 584 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → (𝑁 ∈ (0..^(2↑𝑚)) ↔ (bits‘𝑁) ⊆ (0..^𝑚))) | 
| 24 | 20, 23 | mpbid 232 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → (bits‘𝑁) ⊆ (0..^𝑚)) | 
| 25 | ssfi 9214 | . . 3 ⊢ (((0..^𝑚) ∈ Fin ∧ (bits‘𝑁) ⊆ (0..^𝑚)) → (bits‘𝑁) ∈ Fin) | |
| 26 | 8, 24, 25 | sylancr 587 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → (bits‘𝑁) ∈ Fin) | 
| 27 | 7, 26 | rexlimddv 3160 | 1 ⊢ (𝑁 ∈ ℕ0 → (bits‘𝑁) ∈ Fin) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2107 ∃wrex 3069 ⊆ wss 3950 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 Fincfn 8986 ℝcr 11155 0cc0 11156 1c1 11157 < clt 11296 ℕcn 12267 2c2 12322 ℕ0cn0 12528 ℤcz 12615 ℤ≥cuz 12879 ..^cfzo 13695 ↑cexp 14103 bitscbits 16457 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 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 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-fl 13833 df-seq 14044 df-exp 14104 df-dvds 16292 df-bits 16460 | 
| This theorem is referenced by: bitsinv2 16481 bitsf1ocnv 16482 bitsf1 16484 eulerpartlemgc 34365 eulerpartlemgs2 34383 | 
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