|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > bitsfi | Structured version Visualization version GIF version | ||
| 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 12535 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
| 2 | 2re 12340 | . . . 4 ⊢ 2 ∈ ℝ | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) | 
| 4 | 1lt2 12437 | . . . 4 ⊢ 1 < 2 | |
| 5 | 4 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 1 < 2) | 
| 6 | expnbnd 14271 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ 2 ∈ ℝ ∧ 1 < 2) → ∃𝑚 ∈ ℕ 𝑁 < (2↑𝑚)) | |
| 7 | 1, 3, 5, 6 | syl3anc 1373 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∃𝑚 ∈ ℕ 𝑁 < (2↑𝑚)) | 
| 8 | fzofi 14015 | . . 3 ⊢ (0..^𝑚) ∈ Fin | |
| 9 | simpl 482 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 ∈ ℕ0) | |
| 10 | nn0uz 12920 | . . . . . 6 ⊢ ℕ0 = (ℤ≥‘0) | |
| 11 | 9, 10 | eleqtrdi 2851 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 ∈ (ℤ≥‘0)) | 
| 12 | 2nn 12339 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
| 13 | 12 | a1i 11 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 2 ∈ ℕ) | 
| 14 | simprl 771 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑚 ∈ ℕ) | |
| 15 | 14 | nnnn0d 12587 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑚 ∈ ℕ0) | 
| 16 | 13, 15 | nnexpcld 14284 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → (2↑𝑚) ∈ ℕ) | 
| 17 | 16 | nnzd 12640 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → (2↑𝑚) ∈ ℤ) | 
| 18 | simprr 773 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 < (2↑𝑚)) | |
| 19 | elfzo2 13702 | . . . . 5 ⊢ (𝑁 ∈ (0..^(2↑𝑚)) ↔ (𝑁 ∈ (ℤ≥‘0) ∧ (2↑𝑚) ∈ ℤ ∧ 𝑁 < (2↑𝑚))) | |
| 20 | 11, 17, 18, 19 | syl3anbrc 1344 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 ∈ (0..^(2↑𝑚))) | 
| 21 | 9 | nn0zd 12639 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 ∈ ℤ) | 
| 22 | bitsfzo 16472 | . . . . 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 9213 | . . 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 3161 | 1 ⊢ (𝑁 ∈ ℕ0 → (bits‘𝑁) ∈ Fin) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℝcr 11154 0cc0 11155 1c1 11156 < clt 11295 ℕcn 12266 2c2 12321 ℕ0cn0 12526 ℤcz 12613 ℤ≥cuz 12878 ..^cfzo 13694 ↑cexp 14102 bitscbits 16456 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-dvds 16291 df-bits 16459 | 
| This theorem is referenced by: bitsinv2 16480 bitsf1ocnv 16481 bitsf1 16483 eulerpartlemgc 34364 eulerpartlemgs2 34382 | 
| Copyright terms: Public domain | W3C validator |