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Theorem 3dvdsdec 11307
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 8979 . . . 4  |- ; A B  =  ( (; 1 0  x.  A
)  +  B )
2 9p1e10 8978 . . . . . . . 8  |-  ( 9  +  1 )  = ; 1
0
32eqcomi 2099 . . . . . . 7  |- ; 1 0  =  ( 9  +  1 )
43oveq1i 5700 . . . . . 6  |-  (; 1 0  x.  A
)  =  ( ( 9  +  1 )  x.  A )
5 9cn 8608 . . . . . . 7  |-  9  e.  CC
6 ax-1cn 7535 . . . . . . 7  |-  1  e.  CC
7 3dvdsdec.a . . . . . . . 8  |-  A  e. 
NN0
87nn0cni 8783 . . . . . . 7  |-  A  e.  CC
95, 6, 8adddiri 7596 . . . . . 6  |-  ( ( 9  +  1 )  x.  A )  =  ( ( 9  x.  A )  +  ( 1  x.  A ) )
108mulid2i 7588 . . . . . . 7  |-  ( 1  x.  A )  =  A
1110oveq2i 5701 . . . . . 6  |-  ( ( 9  x.  A )  +  ( 1  x.  A ) )  =  ( ( 9  x.  A )  +  A
)
124, 9, 113eqtri 2119 . . . . 5  |-  (; 1 0  x.  A
)  =  ( ( 9  x.  A )  +  A )
1312oveq1i 5700 . . . 4  |-  ( (; 1
0  x.  A )  +  B )  =  ( ( ( 9  x.  A )  +  A )  +  B
)
145, 8mulcli 7590 . . . . 5  |-  ( 9  x.  A )  e.  CC
15 3dvdsdec.b . . . . . 6  |-  B  e. 
NN0
1615nn0cni 8783 . . . . 5  |-  B  e.  CC
1714, 8, 16addassi 7593 . . . 4  |-  ( ( ( 9  x.  A
)  +  A )  +  B )  =  ( ( 9  x.  A )  +  ( A  +  B ) )
181, 13, 173eqtri 2119 . . 3  |- ; A B  =  ( ( 9  x.  A
)  +  ( A  +  B ) )
1918breq2i 3875 . 2  |-  ( 3 
|| ; A B  <->  3  ||  (
( 9  x.  A
)  +  ( A  +  B ) ) )
20 3z 8877 . . 3  |-  3  e.  ZZ
217nn0zi 8870 . . . 4  |-  A  e.  ZZ
2215nn0zi 8870 . . . 4  |-  B  e.  ZZ
23 zaddcl 8888 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  +  B
)  e.  ZZ )
2421, 22, 23mp2an 418 . . 3  |-  ( A  +  B )  e.  ZZ
25 9nn 8682 . . . . . 6  |-  9  e.  NN
2625nnzi 8869 . . . . 5  |-  9  e.  ZZ
27 zmulcl 8901 . . . . 5  |-  ( ( 9  e.  ZZ  /\  A  e.  ZZ )  ->  ( 9  x.  A
)  e.  ZZ )
2826, 21, 27mp2an 418 . . . 4  |-  ( 9  x.  A )  e.  ZZ
29 zmulcl 8901 . . . . . . 7  |-  ( ( 3  e.  ZZ  /\  A  e.  ZZ )  ->  ( 3  x.  A
)  e.  ZZ )
3020, 21, 29mp2an 418 . . . . . 6  |-  ( 3  x.  A )  e.  ZZ
31 dvdsmul1 11260 . . . . . 6  |-  ( ( 3  e.  ZZ  /\  ( 3  x.  A
)  e.  ZZ )  ->  3  ||  (
3  x.  ( 3  x.  A ) ) )
3220, 30, 31mp2an 418 . . . . 5  |-  3  ||  ( 3  x.  (
3  x.  A ) )
33 3t3e9 8671 . . . . . . . 8  |-  ( 3  x.  3 )  =  9
3433eqcomi 2099 . . . . . . 7  |-  9  =  ( 3  x.  3 )
3534oveq1i 5700 . . . . . 6  |-  ( 9  x.  A )  =  ( ( 3  x.  3 )  x.  A
)
36 3cn 8595 . . . . . . 7  |-  3  e.  CC
3736, 36, 8mulassi 7594 . . . . . 6  |-  ( ( 3  x.  3 )  x.  A )  =  ( 3  x.  (
3  x.  A ) )
3835, 37eqtri 2115 . . . . 5  |-  ( 9  x.  A )  =  ( 3  x.  (
3  x.  A ) )
3932, 38breqtrri 3892 . . . 4  |-  3  ||  ( 9  x.  A
)
4028, 39pm3.2i 267 . . 3  |-  ( ( 9  x.  A )  e.  ZZ  /\  3  ||  ( 9  x.  A
) )
41 dvdsadd2b 11285 . . 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 1280 . 2  |-  ( 3 
||  ( A  +  B )  <->  3  ||  ( ( 9  x.  A )  +  ( A  +  B ) ) )
4319, 42bitr4i 186 1  |-  ( 3 
|| ; A B  <->  3  ||  ( A  +  B )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1445   class class class wbr 3867  (class class class)co 5690   0cc0 7447   1c1 7448    + caddc 7450    x. cmul 7452   3c3 8572   9c9 8578   NN0cn0 8771   ZZcz 8848  ;cdc 8976    || cdvds 11238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-ltadd 7558
This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-5 8582  df-6 8583  df-7 8584  df-8 8585  df-9 8586  df-n0 8772  df-z 8849  df-dec 8977  df-dvds 11239
This theorem is referenced by: (None)
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