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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnpw2pmod | Structured version Visualization version GIF version |
Description: Every positive integer can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020.) |
Ref | Expression |
---|---|
nnpw2pmod | ⊢ (𝑁 ∈ ℕ → 𝑁 = ((2↑((#b‘𝑁) − 1)) + (𝑁 mod (2↑((#b‘𝑁) − 1))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre 11633 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
2 | 2nn 11698 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
3 | 2 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ) |
4 | blennnelnn 44564 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℕ) | |
5 | nnm1nn0 11926 | . . . . . . . 8 ⊢ ((#b‘𝑁) ∈ ℕ → ((#b‘𝑁) − 1) ∈ ℕ0) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((#b‘𝑁) − 1) ∈ ℕ0) |
7 | 3, 6 | nnexpcld 13594 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) ∈ ℕ) |
8 | 7 | nnrpd 12417 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) ∈ ℝ+) |
9 | modeqmodmin 13297 | . . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ (2↑((#b‘𝑁) − 1)) ∈ ℝ+) → (𝑁 mod (2↑((#b‘𝑁) − 1))) = ((𝑁 − (2↑((#b‘𝑁) − 1))) mod (2↑((#b‘𝑁) − 1)))) | |
10 | 1, 8, 9 | syl2anc 584 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 mod (2↑((#b‘𝑁) − 1))) = ((𝑁 − (2↑((#b‘𝑁) − 1))) mod (2↑((#b‘𝑁) − 1)))) |
11 | 7 | nnred 11641 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) ∈ ℝ) |
12 | 1, 11 | resubcld 11056 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − (2↑((#b‘𝑁) − 1))) ∈ ℝ) |
13 | nnpw2blen 44568 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((2↑((#b‘𝑁) − 1)) ≤ 𝑁 ∧ 𝑁 < (2↑(#b‘𝑁)))) | |
14 | 1, 11 | subge0d 11218 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (0 ≤ (𝑁 − (2↑((#b‘𝑁) − 1))) ↔ (2↑((#b‘𝑁) − 1)) ≤ 𝑁)) |
15 | 1, 11, 11 | ltsubadd2d 11226 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑁 − (2↑((#b‘𝑁) − 1))) < (2↑((#b‘𝑁) − 1)) ↔ 𝑁 < ((2↑((#b‘𝑁) − 1)) + (2↑((#b‘𝑁) − 1))))) |
16 | 2cn 11700 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℂ | |
17 | exp1 13423 | . . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ → (2↑1) = 2) | |
18 | 17 | eqcomd 2824 | . . . . . . . . . . . 12 ⊢ (2 ∈ ℂ → 2 = (2↑1)) |
19 | 16, 18 | mp1i 13 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 2 = (2↑1)) |
20 | 19 | oveq1d 7160 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (2 · (2↑((#b‘𝑁) − 1))) = ((2↑1) · (2↑((#b‘𝑁) − 1)))) |
21 | 7 | nncnd 11642 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (2↑((#b‘𝑁) − 1)) ∈ ℂ) |
22 | 21 | 2timesd 11868 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (2 · (2↑((#b‘𝑁) − 1))) = ((2↑((#b‘𝑁) − 1)) + (2↑((#b‘𝑁) − 1)))) |
23 | 16 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) |
24 | 1nn0 11901 | . . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ0 | |
25 | 24 | a1i 11 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℕ0) |
26 | 23, 6, 25 | expaddd 13500 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (2↑(1 + ((#b‘𝑁) − 1))) = ((2↑1) · (2↑((#b‘𝑁) − 1)))) |
27 | 1cnd 10624 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ) | |
28 | 4 | nncnd 11642 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (#b‘𝑁) ∈ ℂ) |
29 | 27, 28 | pncan3d 10988 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (1 + ((#b‘𝑁) − 1)) = (#b‘𝑁)) |
30 | 29 | oveq2d 7161 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (2↑(1 + ((#b‘𝑁) − 1))) = (2↑(#b‘𝑁))) |
31 | 26, 30 | eqtr3d 2855 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((2↑1) · (2↑((#b‘𝑁) − 1))) = (2↑(#b‘𝑁))) |
32 | 20, 22, 31 | 3eqtr3d 2861 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((2↑((#b‘𝑁) − 1)) + (2↑((#b‘𝑁) − 1))) = (2↑(#b‘𝑁))) |
33 | 32 | breq2d 5069 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 < ((2↑((#b‘𝑁) − 1)) + (2↑((#b‘𝑁) − 1))) ↔ 𝑁 < (2↑(#b‘𝑁)))) |
34 | 15, 33 | bitrd 280 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑁 − (2↑((#b‘𝑁) − 1))) < (2↑((#b‘𝑁) − 1)) ↔ 𝑁 < (2↑(#b‘𝑁)))) |
35 | 14, 34 | anbi12d 630 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((0 ≤ (𝑁 − (2↑((#b‘𝑁) − 1))) ∧ (𝑁 − (2↑((#b‘𝑁) − 1))) < (2↑((#b‘𝑁) − 1))) ↔ ((2↑((#b‘𝑁) − 1)) ≤ 𝑁 ∧ 𝑁 < (2↑(#b‘𝑁))))) |
36 | 13, 35 | mpbird 258 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (0 ≤ (𝑁 − (2↑((#b‘𝑁) − 1))) ∧ (𝑁 − (2↑((#b‘𝑁) − 1))) < (2↑((#b‘𝑁) − 1)))) |
37 | modid 13252 | . . . . 5 ⊢ ((((𝑁 − (2↑((#b‘𝑁) − 1))) ∈ ℝ ∧ (2↑((#b‘𝑁) − 1)) ∈ ℝ+) ∧ (0 ≤ (𝑁 − (2↑((#b‘𝑁) − 1))) ∧ (𝑁 − (2↑((#b‘𝑁) − 1))) < (2↑((#b‘𝑁) − 1)))) → ((𝑁 − (2↑((#b‘𝑁) − 1))) mod (2↑((#b‘𝑁) − 1))) = (𝑁 − (2↑((#b‘𝑁) − 1)))) | |
38 | 12, 8, 36, 37 | syl21anc 833 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 − (2↑((#b‘𝑁) − 1))) mod (2↑((#b‘𝑁) − 1))) = (𝑁 − (2↑((#b‘𝑁) − 1)))) |
39 | 10, 38 | eqtr2d 2854 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − (2↑((#b‘𝑁) − 1))) = (𝑁 mod (2↑((#b‘𝑁) − 1)))) |
40 | nncn 11634 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
41 | nnz 11992 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
42 | 41, 7 | zmodcld 13248 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 mod (2↑((#b‘𝑁) − 1))) ∈ ℕ0) |
43 | 42 | nn0cnd 11945 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 mod (2↑((#b‘𝑁) − 1))) ∈ ℂ) |
44 | 40, 21, 43 | subaddd 11003 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝑁 − (2↑((#b‘𝑁) − 1))) = (𝑁 mod (2↑((#b‘𝑁) − 1))) ↔ ((2↑((#b‘𝑁) − 1)) + (𝑁 mod (2↑((#b‘𝑁) − 1)))) = 𝑁)) |
45 | 39, 44 | mpbid 233 | . 2 ⊢ (𝑁 ∈ ℕ → ((2↑((#b‘𝑁) − 1)) + (𝑁 mod (2↑((#b‘𝑁) − 1)))) = 𝑁) |
46 | 45 | eqcomd 2824 | 1 ⊢ (𝑁 ∈ ℕ → 𝑁 = ((2↑((#b‘𝑁) − 1)) + (𝑁 mod (2↑((#b‘𝑁) − 1))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 < clt 10663 ≤ cle 10664 − cmin 10858 ℕcn 11626 2c2 11680 ℕ0cn0 11885 ℝ+crp 12377 mod cmo 13225 ↑cexp 13417 #bcblen 44557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 df-cxp 25068 df-logb 25270 df-blen 44558 |
This theorem is referenced by: nnpw2p 44574 |
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