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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 42391 | . . 3 ⊢ (𝐵 ∈ ℕ → (digit‘𝐵) = (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵))) | |
2 | 1 | 3ad2ant1 1082 | . 2 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → (digit‘𝐵) = (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦ ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵))) |
3 | negeq 10273 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → -𝑘 = -𝐾) | |
4 | 3 | oveq2d 6666 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝐵↑-𝑘) = (𝐵↑-𝐾)) |
5 | 4 | adantr 481 | . . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑟 = 𝑅) → (𝐵↑-𝑘) = (𝐵↑-𝐾)) |
6 | simpr 477 | . . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) | |
7 | 5, 6 | oveq12d 6668 | . . . . 5 ⊢ ((𝑘 = 𝐾 ∧ 𝑟 = 𝑅) → ((𝐵↑-𝑘) · 𝑟) = ((𝐵↑-𝐾) · 𝑅)) |
8 | 7 | fveq2d 6195 | . . . 4 ⊢ ((𝑘 = 𝐾 ∧ 𝑟 = 𝑅) → (⌊‘((𝐵↑-𝑘) · 𝑟)) = (⌊‘((𝐵↑-𝐾) · 𝑅))) |
9 | 8 | oveq1d 6665 | . . 3 ⊢ ((𝑘 = 𝐾 ∧ 𝑟 = 𝑅) → ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵) = ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵)) |
10 | 9 | adantl 482 | . 2 ⊢ (((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) ∧ (𝑘 = 𝐾 ∧ 𝑟 = 𝑅)) → ((⌊‘((𝐵↑-𝑘) · 𝑟)) mod 𝐵) = ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵)) |
11 | simp2 1062 | . 2 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → 𝐾 ∈ ℤ) | |
12 | simp3 1063 | . 2 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → 𝑅 ∈ (0[,)+∞)) | |
13 | ovexd 6680 | . 2 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 𝑅 ∈ (0[,)+∞)) → ((⌊‘((𝐵↑-𝐾) · 𝑅)) mod 𝐵) ∈ V) | |
14 | 2, 10, 11, 12, 13 | ovmpt2d 6788 | 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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 0cc0 9936 · cmul 9941 +∞cpnf 10071 -cneg 10267 ℕcn 11020 ℤcz 11377 [,)cico 12177 ⌊cfl 12591 mod cmo 12668 ↑cexp 12860 digitcdig 42389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-neg 10269 df-z 11378 df-dig 42390 |
This theorem is referenced by: digvalnn0 42393 nn0digval 42394 dignn0fr 42395 dig0 42400 dig2nn0 42405 |
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