![]() |
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 12533 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
2 | 2re 12338 | . . . 4 ⊢ 2 ∈ ℝ | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) |
4 | 1lt2 12435 | . . . 4 ⊢ 1 < 2 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 1 < 2) |
6 | expnbnd 14268 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ 2 ∈ ℝ ∧ 1 < 2) → ∃𝑚 ∈ ℕ 𝑁 < (2↑𝑚)) | |
7 | 1, 3, 5, 6 | syl3anc 1370 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∃𝑚 ∈ ℕ 𝑁 < (2↑𝑚)) |
8 | fzofi 14012 | . . 3 ⊢ (0..^𝑚) ∈ Fin | |
9 | simpl 482 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 ∈ ℕ0) | |
10 | nn0uz 12918 | . . . . . 6 ⊢ ℕ0 = (ℤ≥‘0) | |
11 | 9, 10 | eleqtrdi 2849 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 ∈ (ℤ≥‘0)) |
12 | 2nn 12337 | . . . . . . . 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 12585 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑚 ∈ ℕ0) |
16 | 13, 15 | nnexpcld 14281 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → (2↑𝑚) ∈ ℕ) |
17 | 16 | nnzd 12638 | . . . . 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 13699 | . . . . 5 ⊢ (𝑁 ∈ (0..^(2↑𝑚)) ↔ (𝑁 ∈ (ℤ≥‘0) ∧ (2↑𝑚) ∈ ℤ ∧ 𝑁 < (2↑𝑚))) | |
20 | 11, 17, 18, 19 | syl3anbrc 1342 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 ∈ (0..^(2↑𝑚))) |
21 | 9 | nn0zd 12637 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑚 ∈ ℕ ∧ 𝑁 < (2↑𝑚))) → 𝑁 ∈ ℤ) |
22 | bitsfzo 16469 | . . . . 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 9212 | . . 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 3159 | 1 ⊢ (𝑁 ∈ ℕ0 → (bits‘𝑁) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℝcr 11152 0cc0 11153 1c1 11154 < clt 11293 ℕcn 12264 2c2 12319 ℕ0cn0 12524 ℤcz 12611 ℤ≥cuz 12876 ..^cfzo 13691 ↑cexp 14099 bitscbits 16453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-dvds 16288 df-bits 16456 |
This theorem is referenced by: bitsinv2 16477 bitsf1ocnv 16478 bitsf1 16480 eulerpartlemgc 34344 eulerpartlemgs2 34362 |
Copyright terms: Public domain | W3C validator |