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Mirrors > Home > MPE Home > Th. List > Mathboxes > dig0 | Structured version Visualization version GIF version |
Description: All digits of 0 are 0. (Contributed by AV, 24-May-2020.) |
Ref | Expression |
---|---|
dig0 | ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0e0icopnf 12364 | . . 3 ⊢ 0 ∈ (0[,)+∞) | |
2 | digval 42787 | . . 3 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ ∧ 0 ∈ (0[,)+∞)) → (𝐾(digit‘𝐵)0) = ((⌊‘((𝐵↑-𝐾) · 0)) mod 𝐵)) | |
3 | 1, 2 | mp3an3 1494 | . 2 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = ((⌊‘((𝐵↑-𝐾) · 0)) mod 𝐵)) |
4 | nncn 11109 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ) | |
5 | 4 | adantr 472 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → 𝐵 ∈ ℂ) |
6 | nnne0 11134 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0) | |
7 | 6 | adantr 472 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → 𝐵 ≠ 0) |
8 | znegcl 11493 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → -𝐾 ∈ ℤ) | |
9 | 8 | adantl 473 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → -𝐾 ∈ ℤ) |
10 | 5, 7, 9 | expclzd 13096 | . . . . . . 7 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐵↑-𝐾) ∈ ℂ) |
11 | 10 | mul01d 10316 | . . . . . 6 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → ((𝐵↑-𝐾) · 0) = 0) |
12 | 11 | fveq2d 6276 | . . . . 5 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (⌊‘((𝐵↑-𝐾) · 0)) = (⌊‘0)) |
13 | 0zd 11470 | . . . . . 6 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → 0 ∈ ℤ) | |
14 | flid 12692 | . . . . . 6 ⊢ (0 ∈ ℤ → (⌊‘0) = 0) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (⌊‘0) = 0) |
16 | 12, 15 | eqtrd 2726 | . . . 4 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (⌊‘((𝐵↑-𝐾) · 0)) = 0) |
17 | 16 | oveq1d 6748 | . . 3 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → ((⌊‘((𝐵↑-𝐾) · 0)) mod 𝐵) = (0 mod 𝐵)) |
18 | nnrp 11924 | . . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ+) | |
19 | 0mod 12784 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → (0 mod 𝐵) = 0) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ (𝐵 ∈ ℕ → (0 mod 𝐵) = 0) |
21 | 20 | adantr 472 | . . 3 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (0 mod 𝐵) = 0) |
22 | 17, 21 | eqtrd 2726 | . 2 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → ((⌊‘((𝐵↑-𝐾) · 0)) mod 𝐵) = 0) |
23 | 3, 22 | eqtrd 2726 | 1 ⊢ ((𝐵 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (𝐾(digit‘𝐵)0) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1564 ∈ wcel 2071 ≠ wne 2864 ‘cfv 5969 (class class class)co 6733 ℂcc 10015 0cc0 10017 · cmul 10022 +∞cpnf 10152 -cneg 10348 ℕcn 11101 ℤcz 11458 ℝ+crp 11914 [,)cico 12259 ⌊cfl 12674 mod cmo 12751 ↑cexp 12943 digitcdig 42784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-rep 4847 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 ax-cnex 10073 ax-resscn 10074 ax-1cn 10075 ax-icn 10076 ax-addcl 10077 ax-addrcl 10078 ax-mulcl 10079 ax-mulrcl 10080 ax-mulcom 10081 ax-addass 10082 ax-mulass 10083 ax-distr 10084 ax-i2m1 10085 ax-1ne0 10086 ax-1rid 10087 ax-rnegex 10088 ax-rrecex 10089 ax-cnre 10090 ax-pre-lttri 10091 ax-pre-lttrn 10092 ax-pre-ltadd 10093 ax-pre-mulgt0 10094 ax-pre-sup 10095 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-nel 2968 df-ral 2987 df-rex 2988 df-reu 2989 df-rmo 2990 df-rab 2991 df-v 3274 df-sbc 3510 df-csb 3608 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-pss 3664 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-tp 4258 df-op 4260 df-uni 4513 df-iun 4598 df-br 4729 df-opab 4789 df-mpt 4806 df-tr 4829 df-id 5096 df-eprel 5101 df-po 5107 df-so 5108 df-fr 5145 df-we 5147 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-pred 5761 df-ord 5807 df-on 5808 df-lim 5809 df-suc 5810 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-f1 5974 df-fo 5975 df-f1o 5976 df-fv 5977 df-riota 6694 df-ov 6736 df-oprab 6737 df-mpt2 6738 df-om 7151 df-1st 7253 df-2nd 7254 df-wrecs 7495 df-recs 7556 df-rdg 7594 df-er 7830 df-en 8041 df-dom 8042 df-sdom 8043 df-sup 8432 df-inf 8433 df-pnf 10157 df-mnf 10158 df-xr 10159 df-ltxr 10160 df-le 10161 df-sub 10349 df-neg 10350 df-div 10766 df-nn 11102 df-n0 11374 df-z 11459 df-uz 11769 df-rp 11915 df-ico 12263 df-fl 12676 df-mod 12752 df-seq 12885 df-exp 12944 df-dig 42785 |
This theorem is referenced by: 0dig2pr01 42799 nn0sumshdiglem1 42810 |
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