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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 9642 | . . . 4 ;; ; ; |
4 | sq10e99m1 9641 | . . . . . . . 8 ; ; | |
5 | 4 | oveq1i 5542 | . . . . . . 7 ; ; |
6 | 9nn0 8312 | . . . . . . . . . 10 | |
7 | 6, 6 | deccl 8491 | . . . . . . . . 9 ; |
8 | 7 | nn0cni 8300 | . . . . . . . 8 ; |
9 | ax-1cn 7069 | . . . . . . . 8 | |
10 | 1 | nn0cni 8300 | . . . . . . . 8 |
11 | 8, 9, 10 | adddiri 7130 | . . . . . . 7 ; ; |
12 | 10 | mulid2i 7122 | . . . . . . . 8 |
13 | 12 | oveq2i 5543 | . . . . . . 7 ; ; |
14 | 5, 11, 13 | 3eqtri 2105 | . . . . . 6 ; ; |
15 | 9p1e10 8479 | . . . . . . . . 9 ; | |
16 | 15 | eqcomi 2085 | . . . . . . . 8 ; |
17 | 16 | oveq1i 5542 | . . . . . . 7 ; |
18 | 9cn 8127 | . . . . . . . 8 | |
19 | 2 | nn0cni 8300 | . . . . . . . 8 |
20 | 18, 9, 19 | adddiri 7130 | . . . . . . 7 |
21 | 19 | mulid2i 7122 | . . . . . . . 8 |
22 | 21 | oveq2i 5543 | . . . . . . 7 |
23 | 17, 20, 22 | 3eqtri 2105 | . . . . . 6 ; |
24 | 14, 23 | oveq12i 5544 | . . . . 5 ; ; ; |
25 | 24 | oveq1i 5542 | . . . 4 ; ; ; |
26 | 8, 10 | mulcli 7124 | . . . . . 6 ; |
27 | 18, 19 | mulcli 7124 | . . . . . 6 |
28 | add4 7269 | . . . . . . 7 ; ; ; | |
29 | 28 | oveq1d 5547 | . . . . . 6 ; ; ; |
30 | 26, 10, 27, 19, 29 | mp4an 417 | . . . . 5 ; ; |
31 | 26, 27 | addcli 7123 | . . . . . 6 ; |
32 | 10, 19 | addcli 7123 | . . . . . 6 |
33 | 3dvds2dec.c | . . . . . . 7 | |
34 | 33 | nn0cni 8300 | . . . . . 6 |
35 | 31, 32, 34 | addassi 7127 | . . . . 5 ; ; |
36 | 9t11e99 8606 | . . . . . . . . . . 11 ; ; | |
37 | 36 | eqcomi 2085 | . . . . . . . . . 10 ; ; |
38 | 37 | oveq1i 5542 | . . . . . . . . 9 ; ; |
39 | 1nn0 8304 | . . . . . . . . . . . 12 | |
40 | 39, 39 | deccl 8491 | . . . . . . . . . . 11 ; |
41 | 40 | nn0cni 8300 | . . . . . . . . . 10 ; |
42 | 18, 41, 10 | mulassi 7128 | . . . . . . . . 9 ; ; |
43 | 38, 42 | eqtri 2101 | . . . . . . . 8 ; ; |
44 | 43 | oveq1i 5542 | . . . . . . 7 ; ; |
45 | 41, 10 | mulcli 7124 | . . . . . . . . 9 ; |
46 | 18, 45, 19 | adddii 7129 | . . . . . . . 8 ; ; |
47 | 46 | eqcomi 2085 | . . . . . . 7 ; ; |
48 | 3t3e9 8189 | . . . . . . . . . 10 | |
49 | 48 | eqcomi 2085 | . . . . . . . . 9 |
50 | 49 | oveq1i 5542 | . . . . . . . 8 ; ; |
51 | 3cn 8114 | . . . . . . . . 9 | |
52 | 45, 19 | addcli 7123 | . . . . . . . . 9 ; |
53 | 51, 51, 52 | mulassi 7128 | . . . . . . . 8 ; ; |
54 | 50, 53 | eqtri 2101 | . . . . . . 7 ; ; |
55 | 44, 47, 54 | 3eqtri 2105 | . . . . . 6 ; ; |
56 | 55 | oveq1i 5542 | . . . . 5 ; ; |
57 | 30, 35, 56 | 3eqtri 2105 | . . . 4 ; ; |
58 | 3, 25, 57 | 3eqtri 2105 | . . 3 ;; ; |
59 | 58 | breq2i 3793 | . 2 ;; ; |
60 | 3z 8380 | . . 3 | |
61 | 1 | nn0zi 8373 | . . . . 5 |
62 | 2 | nn0zi 8373 | . . . . 5 |
63 | zaddcl 8391 | . . . . 5 | |
64 | 61, 62, 63 | mp2an 416 | . . . 4 |
65 | 33 | nn0zi 8373 | . . . 4 |
66 | zaddcl 8391 | . . . 4 | |
67 | 64, 65, 66 | mp2an 416 | . . 3 |
68 | 40 | nn0zi 8373 | . . . . . . . 8 ; |
69 | zmulcl 8404 | . . . . . . . 8 ; ; | |
70 | 68, 61, 69 | mp2an 416 | . . . . . . 7 ; |
71 | zaddcl 8391 | . . . . . . 7 ; ; | |
72 | 70, 62, 71 | mp2an 416 | . . . . . 6 ; |
73 | zmulcl 8404 | . . . . . 6 ; ; | |
74 | 60, 72, 73 | mp2an 416 | . . . . 5 ; |
75 | zmulcl 8404 | . . . . 5 ; ; | |
76 | 60, 74, 75 | mp2an 416 | . . . 4 ; |
77 | dvdsmul1 10217 | . . . . 5 ; ; | |
78 | 60, 74, 77 | mp2an 416 | . . . 4 ; |
79 | 76, 78 | pm3.2i 266 | . . 3 ; ; |
80 | dvdsadd2b 10242 | . . 3 ; ; ; | |
81 | 60, 67, 79, 80 | mp3an 1268 | . 2 ; |
82 | 59, 81 | bitr4i 185 | 1 ;; |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 wceq 1284 wcel 1433 class class class wbr 3785 (class class class)co 5532 cc 6979 cc0 6981 c1 6982 caddc 6984 cmul 6986 c2 8089 c3 8090 c9 8096 cn0 8288 cz 8351 ;cdc 8477 cexp 9475 cdvds 10195 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-5 8101 df-6 8102 df-7 8103 df-8 8104 df-9 8105 df-n0 8289 df-z 8352 df-dec 8478 df-uz 8620 df-iseq 9432 df-iexp 9476 df-dvds 10196 |
This theorem is referenced by: (None) |
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