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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 11726 | . 2 ⊢ 5 ∈ ℕ | |
2 | 2nn0 11917 | . . 3 ⊢ 2 ∈ ℕ0 | |
3 | dec5dvds.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
4 | 2, 3 | nn0mulcli 11938 | . 2 ⊢ (2 · 𝐴) ∈ ℕ0 |
5 | dec5dvds.2 | . 2 ⊢ 𝐵 ∈ ℕ | |
6 | 5cn 11728 | . . . . . 6 ⊢ 5 ∈ ℂ | |
7 | 2cn 11715 | . . . . . 6 ⊢ 2 ∈ ℂ | |
8 | 3 | nn0cni 11912 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
9 | 6, 7, 8 | mulassi 10654 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (5 · (2 · 𝐴)) |
10 | 5t2e10 12201 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
11 | 10 | oveq1i 7168 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (;10 · 𝐴) |
12 | 9, 11 | eqtr3i 2848 | . . . 4 ⊢ (5 · (2 · 𝐴)) = (;10 · 𝐴) |
13 | 12 | oveq1i 7168 | . . 3 ⊢ ((5 · (2 · 𝐴)) + 𝐵) = ((;10 · 𝐴) + 𝐵) |
14 | dfdec10 12104 | . . 3 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
15 | 13, 14 | eqtr4i 2849 | . 2 ⊢ ((5 · (2 · 𝐴)) + 𝐵) = ;𝐴𝐵 |
16 | dec5dvds.3 | . 2 ⊢ 𝐵 < 5 | |
17 | 1, 4, 5, 15, 16 | ndvdsi 15765 | 1 ⊢ ¬ 5 ∥ ;𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 < clt 10677 ℕcn 11640 2c2 11695 5c5 11698 ℕ0cn0 11900 ;cdc 12101 ∥ cdvds 15609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 |
This theorem is referenced by: dec5dvds2 16403 43prm 16457 83prm 16458 163prm 16460 631prm 16462 31prm 43767 |
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