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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > digval | Structured version Visualization version GIF version |
Description: The 𝐾 th digit of a nonnegative real number 𝑅 in the positional system with base 𝐵. (Contributed by AV, 23-May-2020.) |
Ref | Expression |
---|---|
digval | ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | digfval 45011 | . . 3 ⊢ (𝐵 ∈ ℕ → (digit‘𝐵) = (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵))) | |
2 | 1 | 3ad2ant1 1130 | . 2 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (digit‘𝐵) = (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵))) |
3 | negeq 10867 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → -𝑘 = -𝐾) | |
4 | 3 | oveq2d 7151 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝐵↑-𝑘) = (𝐵↑-𝐾)) |
5 | 4 | adantr 484 | . . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑟 = 𝑅) → (𝐵↑-𝑘) = (𝐵↑-𝐾)) |
6 | simpr 488 | . . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) | |
7 | 5, 6 | oveq12d 7153 | . . . . 5 ⊢ ((𝑘 = 𝐾 ∧ 𝑟 = 𝑅) → ((𝐵↑-𝑘) · 𝑟) = ((𝐵↑-𝐾) · 𝑅)) |
8 | 7 | fveq2d 6649 | . . . 4 ⊢ ((𝑘 = 𝐾 ∧ 𝑟 = 𝑅) → (⌊‘((𝐵↑-𝑘) · 𝑟)) = (⌊‘((𝐵↑-𝐾) · 𝑅))) |
9 | 8 | oveq1d 7150 | . . 3 ⊢ ((𝑘 = 𝐾 ∧ 𝑟 = 𝑅) → ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵) = ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵)) |
10 | 9 | adantl 485 | . 2 ⊢ (((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) ∧ (𝑘 = 𝐾 ∧ 𝑟 = 𝑅)) → ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵) = ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵)) |
11 | simp2 1134 | . 2 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → 𝐾 ∈ ℤ) | |
12 | simp3 1135 | . 2 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → 𝑅 ∈ (0[,)+∞)) | |
13 | ovexd 7170 | . 2 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵) ∈ V) | |
14 | 2, 10, 11, 12, 13 | ovmpod 7281 | 1 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)𝑅) = ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 0cc0 10526 · cmul 10531 +∞cpnf 10661 -cneg 10860 ℕcn 11625 ℤcz 11969 [,)cico 12728 ⌊cfl 13155 mod cmo 13232 ↑cexp 13425 digitcdig 45009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-neg 10862 df-z 11970 df-dig 45010 |
This theorem is referenced by: digvalnn0 45013 nn0digval 45014 dignn0fr 45015 dig0 45020 dig2nn0 45025 |
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