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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 11711 | . 2 ⊢ 5 ∈ ℕ | |
2 | 2nn0 11902 | . . 3 ⊢ 2 ∈ ℕ0 | |
3 | dec5dvds.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
4 | 2, 3 | nn0mulcli 11923 | . 2 ⊢ (2 · 𝐴) ∈ ℕ0 |
5 | dec5dvds.2 | . 2 ⊢ 𝐵 ∈ ℕ | |
6 | 5cn 11713 | . . . . . 6 ⊢ 5 ∈ ℂ | |
7 | 2cn 11700 | . . . . . 6 ⊢ 2 ∈ ℂ | |
8 | 3 | nn0cni 11897 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
9 | 6, 7, 8 | mulassi 10640 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (5 · (2 · 𝐴)) |
10 | 5t2e10 12186 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
11 | 10 | oveq1i 7155 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (;10 · 𝐴) |
12 | 9, 11 | eqtr3i 2843 | . . . 4 ⊢ (5 · (2 · 𝐴)) = (;10 · 𝐴) |
13 | 12 | oveq1i 7155 | . . 3 ⊢ ((5 · (2 · 𝐴)) + 𝐵) = ((;10 · 𝐴) + 𝐵) |
14 | dfdec10 12089 | . . 3 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
15 | 13, 14 | eqtr4i 2844 | . 2 ⊢ ((5 · (2 · 𝐴)) + 𝐵) = ;𝐴𝐵 |
16 | dec5dvds.3 | . 2 ⊢ 𝐵 < 5 | |
17 | 1, 4, 5, 15, 16 | ndvdsi 15751 | 1 ⊢ ¬ 5 ∥ ;𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 < clt 10663 ℕcn 11626 2c2 11680 5c5 11683 ℕ0cn0 11885 ;cdc 12086 ∥ cdvds 15595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 |
This theorem is referenced by: dec5dvds2 16389 43prm 16443 83prm 16444 163prm 16446 631prm 16448 31prm 43637 |
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