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Mirrors > Home > MPE Home > Th. List > dec5dvds | Structured version Visualization version GIF version |
Description: Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
dec5dvds.1 | ⊢ 𝐴 ∈ ℕ0 |
dec5dvds.2 | ⊢ 𝐵 ∈ ℕ |
dec5dvds.3 | ⊢ 𝐵 < 5 |
Ref | Expression |
---|---|
dec5dvds | ⊢ ¬ 5 ∥ ;𝐴𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn 12172 | . 2 ⊢ 5 ∈ ℕ | |
2 | 2nn0 12363 | . . 3 ⊢ 2 ∈ ℕ0 | |
3 | dec5dvds.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
4 | 2, 3 | nn0mulcli 12384 | . 2 ⊢ (2 · 𝐴) ∈ ℕ0 |
5 | dec5dvds.2 | . 2 ⊢ 𝐵 ∈ ℕ | |
6 | 5cn 12174 | . . . . . 6 ⊢ 5 ∈ ℂ | |
7 | 2cn 12161 | . . . . . 6 ⊢ 2 ∈ ℂ | |
8 | 3 | nn0cni 12358 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
9 | 6, 7, 8 | mulassi 11099 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (5 · (2 · 𝐴)) |
10 | 5t2e10 12650 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
11 | 10 | oveq1i 7359 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (;10 · 𝐴) |
12 | 9, 11 | eqtr3i 2767 | . . . 4 ⊢ (5 · (2 · 𝐴)) = (;10 · 𝐴) |
13 | 12 | oveq1i 7359 | . . 3 ⊢ ((5 · (2 · 𝐴)) + 𝐵) = ((;10 · 𝐴) + 𝐵) |
14 | dfdec10 12553 | . . 3 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
15 | 13, 14 | eqtr4i 2768 | . 2 ⊢ ((5 · (2 · 𝐴)) + 𝐵) = ;𝐴𝐵 |
16 | dec5dvds.3 | . 2 ⊢ 𝐵 < 5 | |
17 | 1, 4, 5, 15, 16 | ndvdsi 16228 | 1 ⊢ ¬ 5 ∥ ;𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2106 class class class wbr 5103 (class class class)co 7349 0cc0 10984 1c1 10985 + caddc 10987 · cmul 10989 < clt 11122 ℕcn 12086 2c2 12141 5c5 12144 ℕ0cn0 12346 ;cdc 12550 ∥ cdvds 16070 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 ax-pre-sup 11062 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-1st 7911 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-sup 9311 df-inf 9312 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-div 11746 df-nn 12087 df-2 12149 df-3 12150 df-4 12151 df-5 12152 df-6 12153 df-7 12154 df-8 12155 df-9 12156 df-n0 12347 df-z 12433 df-dec 12551 df-uz 12696 df-rp 12844 df-fz 13353 df-seq 13835 df-exp 13896 df-cj 14917 df-re 14918 df-im 14919 df-sqrt 15053 df-abs 15054 df-dvds 16071 |
This theorem is referenced by: dec5dvds2 16871 43prm 16928 83prm 16929 163prm 16931 631prm 16933 31prm 45538 |
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