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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 11065 | . 2 ⊢ 5 ∈ ℕ | |
2 | 2nn0 11186 | . . 3 ⊢ 2 ∈ ℕ0 | |
3 | dec5dvds.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
4 | 2, 3 | nn0mulcli 11208 | . 2 ⊢ (2 · 𝐴) ∈ ℕ0 |
5 | dec5dvds.2 | . 2 ⊢ 𝐵 ∈ ℕ | |
6 | 5cn 10977 | . . . . . 6 ⊢ 5 ∈ ℂ | |
7 | 2cn 10968 | . . . . . 6 ⊢ 2 ∈ ℂ | |
8 | 3 | nn0cni 11181 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
9 | 6, 7, 8 | mulassi 9928 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (5 · (2 · 𝐴)) |
10 | 5t2e10 11510 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
11 | 10 | oveq1i 6559 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (;10 · 𝐴) |
12 | 9, 11 | eqtr3i 2634 | . . . 4 ⊢ (5 · (2 · 𝐴)) = (;10 · 𝐴) |
13 | 12 | oveq1i 6559 | . . 3 ⊢ ((5 · (2 · 𝐴)) + 𝐵) = ((;10 · 𝐴) + 𝐵) |
14 | dfdec10 11373 | . . 3 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
15 | 13, 14 | eqtr4i 2635 | . 2 ⊢ ((5 · (2 · 𝐴)) + 𝐵) = ;𝐴𝐵 |
16 | dec5dvds.3 | . 2 ⊢ 𝐵 < 5 | |
17 | 1, 4, 5, 15, 16 | ndvdsi 14974 | 1 ⊢ ¬ 5 ∥ ;𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ℕcn 10897 2c2 10947 5c5 10950 ℕ0cn0 11169 ;cdc 11369 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 |
This theorem is referenced by: dec5dvds2 15607 43prm 15667 83prm 15668 163prm 15670 631prm 15672 31prm 40050 |
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