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Mirrors > Home > ILE Home > Th. List > 3dvds2dec | Unicode version |
Description: A decimal number is divisible by three iff the sum of its three "digits" is divisible by three. The term "digits" in its narrow sense is only correct if , and actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers , and . (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
Ref | Expression |
---|---|
3dvdsdec.a | |
3dvdsdec.b | |
3dvds2dec.c |
Ref | Expression |
---|---|
3dvds2dec | ;; |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3dvdsdec.a | . . . . 5 | |
2 | 3dvdsdec.b | . . . . 5 | |
3 | 1, 2 | 3dec 10468 | . . . 4 ;; ; ; |
4 | sq10e99m1 10467 | . . . . . . . 8 ; ; | |
5 | 4 | oveq1i 5784 | . . . . . . 7 ; ; |
6 | 9nn0 9008 | . . . . . . . . . 10 | |
7 | 6, 6 | deccl 9203 | . . . . . . . . 9 ; |
8 | 7 | nn0cni 8996 | . . . . . . . 8 ; |
9 | ax-1cn 7720 | . . . . . . . 8 | |
10 | 1 | nn0cni 8996 | . . . . . . . 8 |
11 | 8, 9, 10 | adddiri 7784 | . . . . . . 7 ; ; |
12 | 10 | mulid2i 7776 | . . . . . . . 8 |
13 | 12 | oveq2i 5785 | . . . . . . 7 ; ; |
14 | 5, 11, 13 | 3eqtri 2164 | . . . . . 6 ; ; |
15 | 9p1e10 9191 | . . . . . . . . 9 ; | |
16 | 15 | eqcomi 2143 | . . . . . . . 8 ; |
17 | 16 | oveq1i 5784 | . . . . . . 7 ; |
18 | 9cn 8815 | . . . . . . . 8 | |
19 | 2 | nn0cni 8996 | . . . . . . . 8 |
20 | 18, 9, 19 | adddiri 7784 | . . . . . . 7 |
21 | 19 | mulid2i 7776 | . . . . . . . 8 |
22 | 21 | oveq2i 5785 | . . . . . . 7 |
23 | 17, 20, 22 | 3eqtri 2164 | . . . . . 6 ; |
24 | 14, 23 | oveq12i 5786 | . . . . 5 ; ; ; |
25 | 24 | oveq1i 5784 | . . . 4 ; ; ; |
26 | 8, 10 | mulcli 7778 | . . . . . 6 ; |
27 | 18, 19 | mulcli 7778 | . . . . . 6 |
28 | add4 7930 | . . . . . . 7 ; ; ; | |
29 | 28 | oveq1d 5789 | . . . . . 6 ; ; ; |
30 | 26, 10, 27, 19, 29 | mp4an 423 | . . . . 5 ; ; |
31 | 26, 27 | addcli 7777 | . . . . . 6 ; |
32 | 10, 19 | addcli 7777 | . . . . . 6 |
33 | 3dvds2dec.c | . . . . . . 7 | |
34 | 33 | nn0cni 8996 | . . . . . 6 |
35 | 31, 32, 34 | addassi 7781 | . . . . 5 ; ; |
36 | 9t11e99 9318 | . . . . . . . . . . 11 ; ; | |
37 | 36 | eqcomi 2143 | . . . . . . . . . 10 ; ; |
38 | 37 | oveq1i 5784 | . . . . . . . . 9 ; ; |
39 | 1nn0 9000 | . . . . . . . . . . . 12 | |
40 | 39, 39 | deccl 9203 | . . . . . . . . . . 11 ; |
41 | 40 | nn0cni 8996 | . . . . . . . . . 10 ; |
42 | 18, 41, 10 | mulassi 7782 | . . . . . . . . 9 ; ; |
43 | 38, 42 | eqtri 2160 | . . . . . . . 8 ; ; |
44 | 43 | oveq1i 5784 | . . . . . . 7 ; ; |
45 | 41, 10 | mulcli 7778 | . . . . . . . . 9 ; |
46 | 18, 45, 19 | adddii 7783 | . . . . . . . 8 ; ; |
47 | 46 | eqcomi 2143 | . . . . . . 7 ; ; |
48 | 3t3e9 8884 | . . . . . . . . . 10 | |
49 | 48 | eqcomi 2143 | . . . . . . . . 9 |
50 | 49 | oveq1i 5784 | . . . . . . . 8 ; ; |
51 | 3cn 8802 | . . . . . . . . 9 | |
52 | 45, 19 | addcli 7777 | . . . . . . . . 9 ; |
53 | 51, 51, 52 | mulassi 7782 | . . . . . . . 8 ; ; |
54 | 50, 53 | eqtri 2160 | . . . . . . 7 ; ; |
55 | 44, 47, 54 | 3eqtri 2164 | . . . . . 6 ; ; |
56 | 55 | oveq1i 5784 | . . . . 5 ; ; |
57 | 30, 35, 56 | 3eqtri 2164 | . . . 4 ; ; |
58 | 3, 25, 57 | 3eqtri 2164 | . . 3 ;; ; |
59 | 58 | breq2i 3937 | . 2 ;; ; |
60 | 3z 9090 | . . 3 | |
61 | 1 | nn0zi 9083 | . . . . 5 |
62 | 2 | nn0zi 9083 | . . . . 5 |
63 | zaddcl 9101 | . . . . 5 | |
64 | 61, 62, 63 | mp2an 422 | . . . 4 |
65 | 33 | nn0zi 9083 | . . . 4 |
66 | zaddcl 9101 | . . . 4 | |
67 | 64, 65, 66 | mp2an 422 | . . 3 |
68 | 40 | nn0zi 9083 | . . . . . . . 8 ; |
69 | zmulcl 9114 | . . . . . . . 8 ; ; | |
70 | 68, 61, 69 | mp2an 422 | . . . . . . 7 ; |
71 | zaddcl 9101 | . . . . . . 7 ; ; | |
72 | 70, 62, 71 | mp2an 422 | . . . . . 6 ; |
73 | zmulcl 9114 | . . . . . 6 ; ; | |
74 | 60, 72, 73 | mp2an 422 | . . . . 5 ; |
75 | zmulcl 9114 | . . . . 5 ; ; | |
76 | 60, 74, 75 | mp2an 422 | . . . 4 ; |
77 | dvdsmul1 11522 | . . . . 5 ; ; | |
78 | 60, 74, 77 | mp2an 422 | . . . 4 ; |
79 | 76, 78 | pm3.2i 270 | . . 3 ; ; |
80 | dvdsadd2b 11547 | . . 3 ; ; ; | |
81 | 60, 67, 79, 80 | mp3an 1315 | . 2 ; |
82 | 59, 81 | bitr4i 186 | 1 ;; |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wcel 1480 class class class wbr 3929 (class class class)co 5774 cc 7625 cc0 7627 c1 7628 caddc 7630 cmul 7632 c2 8778 c3 8779 c9 8785 cn0 8984 cz 9061 ;cdc 9189 cexp 10299 cdvds 11500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-mulrcl 7726 ax-addcom 7727 ax-mulcom 7728 ax-addass 7729 ax-mulass 7730 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-1rid 7734 ax-0id 7735 ax-rnegex 7736 ax-precex 7737 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-apti 7742 ax-pre-ltadd 7743 ax-pre-mulgt0 7744 ax-pre-mulext 7745 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-reap 8344 df-ap 8351 df-div 8440 df-inn 8728 df-2 8786 df-3 8787 df-4 8788 df-5 8789 df-6 8790 df-7 8791 df-8 8792 df-9 8793 df-n0 8985 df-z 9062 df-dec 9190 df-uz 9334 df-seqfrec 10226 df-exp 10300 df-dvds 11501 |
This theorem is referenced by: (None) |
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