Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > digvalnn0 | Structured version Visualization version GIF version |
Description: The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵 is a nonnegative integer. (Contributed by AV, 28-May-2020.) |
Ref | Expression |
---|---|
digvalnn0 | ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | digval 44665 | . 2 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵)) | |
2 | nnre 11647 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ) | |
3 | 2 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → 𝐵 ∈ ℝ) |
4 | nnne0 11674 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0) | |
5 | 4 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → 𝐵 ≠ 0) |
6 | znegcl 12020 | . . . . . . 7 ⊢ (𝐾 ∈ ℤ → -𝐾 ∈ ℤ) | |
7 | 6 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → -𝐾 ∈ ℤ) |
8 | 3, 5, 7 | reexpclzd 13613 | . . . . 5 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐵↑-𝐾) ∈ ℝ) |
9 | elrege0 12845 | . . . . . . 7 ⊢ (𝑅 ∈ (0[,)+∞) ↔ (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅)) | |
10 | 9 | simplbi 500 | . . . . . 6 ⊢ (𝑅 ∈ (0[,)+∞) → 𝑅 ∈ ℝ) |
11 | 10 | 3ad2ant3 1131 | . . . . 5 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → 𝑅 ∈ ℝ) |
12 | 8, 11 | remulcld 10673 | . . . 4 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → ((𝐵↑-𝐾) · 𝑅) ∈ ℝ) |
13 | 12 | flcld 13171 | . . 3 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (⌊‘((𝐵↑-𝐾) · 𝑅)) ∈ ℤ) |
14 | simp1 1132 | . . 3 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → 𝐵 ∈ ℕ) | |
15 | 13, 14 | zmodcld 13263 | . 2 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵) ∈ ℕ0) |
16 | 1, 15 | eqeltrd 2915 | 1 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 · cmul 10544 +∞cpnf 10674 ≤ cle 10678 -cneg 10873 ℕcn 11640 ℕ0cn0 11900 ℤcz 11984 [,)cico 12743 ⌊cfl 13163 mod cmo 13240 ↑cexp 13432 digitcdig 44662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-dig 44663 |
This theorem is referenced by: nn0sumshdiglemA 44686 nn0sumshdiglemB 44687 nn0sumshdiglem2 44689 nn0mullong 44692 |
Copyright terms: Public domain | W3C validator |