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Theorem 3dvdsdec 11901
Description: A decimal number is divisible by three iff the sum of its two "digits" is divisible by three. The term "digits" in its narrow sense is only correct if  A and  B actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers  A and  B, especially if  A is itself a decimal number, e.g.,  A  = ; C D. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 8-Sep-2021.)
Hypotheses
Ref Expression
3dvdsdec.a  |-  A  e. 
NN0
3dvdsdec.b  |-  B  e. 
NN0
Assertion
Ref Expression
3dvdsdec  |-  ( 3 
|| ; A B  <->  3  ||  ( A  +  B )
)

Proof of Theorem 3dvdsdec
StepHypRef Expression
1 dfdec10 9416 . . . 4  |- ; A B  =  ( (; 1 0  x.  A
)  +  B )
2 9p1e10 9415 . . . . . . . 8  |-  ( 9  +  1 )  = ; 1
0
32eqcomi 2193 . . . . . . 7  |- ; 1 0  =  ( 9  +  1 )
43oveq1i 5905 . . . . . 6  |-  (; 1 0  x.  A
)  =  ( ( 9  +  1 )  x.  A )
5 9cn 9036 . . . . . . 7  |-  9  e.  CC
6 ax-1cn 7933 . . . . . . 7  |-  1  e.  CC
7 3dvdsdec.a . . . . . . . 8  |-  A  e. 
NN0
87nn0cni 9217 . . . . . . 7  |-  A  e.  CC
95, 6, 8adddiri 7997 . . . . . 6  |-  ( ( 9  +  1 )  x.  A )  =  ( ( 9  x.  A )  +  ( 1  x.  A ) )
108mullidi 7989 . . . . . . 7  |-  ( 1  x.  A )  =  A
1110oveq2i 5906 . . . . . 6  |-  ( ( 9  x.  A )  +  ( 1  x.  A ) )  =  ( ( 9  x.  A )  +  A
)
124, 9, 113eqtri 2214 . . . . 5  |-  (; 1 0  x.  A
)  =  ( ( 9  x.  A )  +  A )
1312oveq1i 5905 . . . 4  |-  ( (; 1
0  x.  A )  +  B )  =  ( ( ( 9  x.  A )  +  A )  +  B
)
145, 8mulcli 7991 . . . . 5  |-  ( 9  x.  A )  e.  CC
15 3dvdsdec.b . . . . . 6  |-  B  e. 
NN0
1615nn0cni 9217 . . . . 5  |-  B  e.  CC
1714, 8, 16addassi 7994 . . . 4  |-  ( ( ( 9  x.  A
)  +  A )  +  B )  =  ( ( 9  x.  A )  +  ( A  +  B ) )
181, 13, 173eqtri 2214 . . 3  |- ; A B  =  ( ( 9  x.  A
)  +  ( A  +  B ) )
1918breq2i 4026 . 2  |-  ( 3 
|| ; A B  <->  3  ||  (
( 9  x.  A
)  +  ( A  +  B ) ) )
20 3z 9311 . . 3  |-  3  e.  ZZ
217nn0zi 9304 . . . 4  |-  A  e.  ZZ
2215nn0zi 9304 . . . 4  |-  B  e.  ZZ
23 zaddcl 9322 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  +  B
)  e.  ZZ )
2421, 22, 23mp2an 426 . . 3  |-  ( A  +  B )  e.  ZZ
25 9nn 9116 . . . . . 6  |-  9  e.  NN
2625nnzi 9303 . . . . 5  |-  9  e.  ZZ
27 zmulcl 9335 . . . . 5  |-  ( ( 9  e.  ZZ  /\  A  e.  ZZ )  ->  ( 9  x.  A
)  e.  ZZ )
2826, 21, 27mp2an 426 . . . 4  |-  ( 9  x.  A )  e.  ZZ
29 zmulcl 9335 . . . . . . 7  |-  ( ( 3  e.  ZZ  /\  A  e.  ZZ )  ->  ( 3  x.  A
)  e.  ZZ )
3020, 21, 29mp2an 426 . . . . . 6  |-  ( 3  x.  A )  e.  ZZ
31 dvdsmul1 11851 . . . . . 6  |-  ( ( 3  e.  ZZ  /\  ( 3  x.  A
)  e.  ZZ )  ->  3  ||  (
3  x.  ( 3  x.  A ) ) )
3220, 30, 31mp2an 426 . . . . 5  |-  3  ||  ( 3  x.  (
3  x.  A ) )
33 3t3e9 9105 . . . . . . . 8  |-  ( 3  x.  3 )  =  9
3433eqcomi 2193 . . . . . . 7  |-  9  =  ( 3  x.  3 )
3534oveq1i 5905 . . . . . 6  |-  ( 9  x.  A )  =  ( ( 3  x.  3 )  x.  A
)
36 3cn 9023 . . . . . . 7  |-  3  e.  CC
3736, 36, 8mulassi 7995 . . . . . 6  |-  ( ( 3  x.  3 )  x.  A )  =  ( 3  x.  (
3  x.  A ) )
3835, 37eqtri 2210 . . . . 5  |-  ( 9  x.  A )  =  ( 3  x.  (
3  x.  A ) )
3932, 38breqtrri 4045 . . . 4  |-  3  ||  ( 9  x.  A
)
4028, 39pm3.2i 272 . . 3  |-  ( ( 9  x.  A )  e.  ZZ  /\  3  ||  ( 9  x.  A
) )
41 dvdsadd2b 11878 . . 3  |-  ( ( 3  e.  ZZ  /\  ( A  +  B
)  e.  ZZ  /\  ( ( 9  x.  A )  e.  ZZ  /\  3  ||  ( 9  x.  A ) ) )  ->  ( 3 
||  ( A  +  B )  <->  3  ||  ( ( 9  x.  A )  +  ( A  +  B ) ) ) )
4220, 24, 40, 41mp3an 1348 . 2  |-  ( 3 
||  ( A  +  B )  <->  3  ||  ( ( 9  x.  A )  +  ( A  +  B ) ) )
4319, 42bitr4i 187 1  |-  ( 3 
|| ; A B  <->  3  ||  ( A  +  B )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    e. wcel 2160   class class class wbr 4018  (class class class)co 5895   0cc0 7840   1c1 7841    + caddc 7843    x. cmul 7845   3c3 9000   9c9 9006   NN0cn0 9205   ZZcz 9282  ;cdc 9413    || cdvds 11825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-ltadd 7956
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-5 9010  df-6 9011  df-7 9012  df-8 9013  df-9 9014  df-n0 9206  df-z 9283  df-dec 9414  df-dvds 11826
This theorem is referenced by: (None)
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