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Theorem 3dvdsdec 11872
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 9389 . . . 4  |- ; A B  =  ( (; 1 0  x.  A
)  +  B )
2 9p1e10 9388 . . . . . . . 8  |-  ( 9  +  1 )  = ; 1
0
32eqcomi 2181 . . . . . . 7  |- ; 1 0  =  ( 9  +  1 )
43oveq1i 5887 . . . . . 6  |-  (; 1 0  x.  A
)  =  ( ( 9  +  1 )  x.  A )
5 9cn 9009 . . . . . . 7  |-  9  e.  CC
6 ax-1cn 7906 . . . . . . 7  |-  1  e.  CC
7 3dvdsdec.a . . . . . . . 8  |-  A  e. 
NN0
87nn0cni 9190 . . . . . . 7  |-  A  e.  CC
95, 6, 8adddiri 7970 . . . . . 6  |-  ( ( 9  +  1 )  x.  A )  =  ( ( 9  x.  A )  +  ( 1  x.  A ) )
108mullidi 7962 . . . . . . 7  |-  ( 1  x.  A )  =  A
1110oveq2i 5888 . . . . . 6  |-  ( ( 9  x.  A )  +  ( 1  x.  A ) )  =  ( ( 9  x.  A )  +  A
)
124, 9, 113eqtri 2202 . . . . 5  |-  (; 1 0  x.  A
)  =  ( ( 9  x.  A )  +  A )
1312oveq1i 5887 . . . 4  |-  ( (; 1
0  x.  A )  +  B )  =  ( ( ( 9  x.  A )  +  A )  +  B
)
145, 8mulcli 7964 . . . . 5  |-  ( 9  x.  A )  e.  CC
15 3dvdsdec.b . . . . . 6  |-  B  e. 
NN0
1615nn0cni 9190 . . . . 5  |-  B  e.  CC
1714, 8, 16addassi 7967 . . . 4  |-  ( ( ( 9  x.  A
)  +  A )  +  B )  =  ( ( 9  x.  A )  +  ( A  +  B ) )
181, 13, 173eqtri 2202 . . 3  |- ; A B  =  ( ( 9  x.  A
)  +  ( A  +  B ) )
1918breq2i 4013 . 2  |-  ( 3 
|| ; A B  <->  3  ||  (
( 9  x.  A
)  +  ( A  +  B ) ) )
20 3z 9284 . . 3  |-  3  e.  ZZ
217nn0zi 9277 . . . 4  |-  A  e.  ZZ
2215nn0zi 9277 . . . 4  |-  B  e.  ZZ
23 zaddcl 9295 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  +  B
)  e.  ZZ )
2421, 22, 23mp2an 426 . . 3  |-  ( A  +  B )  e.  ZZ
25 9nn 9089 . . . . . 6  |-  9  e.  NN
2625nnzi 9276 . . . . 5  |-  9  e.  ZZ
27 zmulcl 9308 . . . . 5  |-  ( ( 9  e.  ZZ  /\  A  e.  ZZ )  ->  ( 9  x.  A
)  e.  ZZ )
2826, 21, 27mp2an 426 . . . 4  |-  ( 9  x.  A )  e.  ZZ
29 zmulcl 9308 . . . . . . 7  |-  ( ( 3  e.  ZZ  /\  A  e.  ZZ )  ->  ( 3  x.  A
)  e.  ZZ )
3020, 21, 29mp2an 426 . . . . . 6  |-  ( 3  x.  A )  e.  ZZ
31 dvdsmul1 11822 . . . . . 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 9078 . . . . . . . 8  |-  ( 3  x.  3 )  =  9
3433eqcomi 2181 . . . . . . 7  |-  9  =  ( 3  x.  3 )
3534oveq1i 5887 . . . . . 6  |-  ( 9  x.  A )  =  ( ( 3  x.  3 )  x.  A
)
36 3cn 8996 . . . . . . 7  |-  3  e.  CC
3736, 36, 8mulassi 7968 . . . . . 6  |-  ( ( 3  x.  3 )  x.  A )  =  ( 3  x.  (
3  x.  A ) )
3835, 37eqtri 2198 . . . . 5  |-  ( 9  x.  A )  =  ( 3  x.  (
3  x.  A ) )
3932, 38breqtrri 4032 . . . 4  |-  3  ||  ( 9  x.  A
)
4028, 39pm3.2i 272 . . 3  |-  ( ( 9  x.  A )  e.  ZZ  /\  3  ||  ( 9  x.  A
) )
41 dvdsadd2b 11849 . . 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 1337 . 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 2148   class class class wbr 4005  (class class class)co 5877   0cc0 7813   1c1 7814    + caddc 7816    x. cmul 7818   3c3 8973   9c9 8979   NN0cn0 9178   ZZcz 9255  ;cdc 9386    || cdvds 11796
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-7 8985  df-8 8986  df-9 8987  df-n0 9179  df-z 9256  df-dec 9387  df-dvds 11797
This theorem is referenced by: (None)
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