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Theorem 3dvds2dec 11205
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  A,  B and  C actually are digits (i.e. nonnegative integers less than 10). However, this theorem holds for arbitrary nonnegative integers  A,  B and  C. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
Hypotheses
Ref Expression
3dvdsdec.a  |-  A  e. 
NN0
3dvdsdec.b  |-  B  e. 
NN0
3dvds2dec.c  |-  C  e. 
NN0
Assertion
Ref Expression
3dvds2dec  |-  ( 3 
|| ;; A B C  <->  3  ||  ( ( A  +  B )  +  C
) )

Proof of Theorem 3dvds2dec
StepHypRef Expression
1 3dvdsdec.a . . . . 5  |-  A  e. 
NN0
2 3dvdsdec.b . . . . 5  |-  B  e. 
NN0
31, 23dec 10184 . . . 4  |- ;; A B C  =  (
( ( (; 1 0 ^ 2 )  x.  A )  +  (; 1 0  x.  B
) )  +  C
)
4 sq10e99m1 10183 . . . . . . . 8  |-  (; 1 0 ^ 2 )  =  (; 9 9  +  1 )
54oveq1i 5676 . . . . . . 7  |-  ( (; 1
0 ^ 2 )  x.  A )  =  ( (; 9 9  +  1 )  x.  A )
6 9nn0 8758 . . . . . . . . . 10  |-  9  e.  NN0
76, 6deccl 8952 . . . . . . . . 9  |- ; 9 9  e.  NN0
87nn0cni 8746 . . . . . . . 8  |- ; 9 9  e.  CC
9 ax-1cn 7499 . . . . . . . 8  |-  1  e.  CC
101nn0cni 8746 . . . . . . . 8  |-  A  e.  CC
118, 9, 10adddiri 7560 . . . . . . 7  |-  ( (; 9
9  +  1 )  x.  A )  =  ( (; 9 9  x.  A
)  +  ( 1  x.  A ) )
1210mulid2i 7552 . . . . . . . 8  |-  ( 1  x.  A )  =  A
1312oveq2i 5677 . . . . . . 7  |-  ( (; 9
9  x.  A )  +  ( 1  x.  A ) )  =  ( (; 9 9  x.  A
)  +  A )
145, 11, 133eqtri 2113 . . . . . 6  |-  ( (; 1
0 ^ 2 )  x.  A )  =  ( (; 9 9  x.  A
)  +  A )
15 9p1e10 8940 . . . . . . . . 9  |-  ( 9  +  1 )  = ; 1
0
1615eqcomi 2093 . . . . . . . 8  |- ; 1 0  =  ( 9  +  1 )
1716oveq1i 5676 . . . . . . 7  |-  (; 1 0  x.  B
)  =  ( ( 9  +  1 )  x.  B )
18 9cn 8571 . . . . . . . 8  |-  9  e.  CC
192nn0cni 8746 . . . . . . . 8  |-  B  e.  CC
2018, 9, 19adddiri 7560 . . . . . . 7  |-  ( ( 9  +  1 )  x.  B )  =  ( ( 9  x.  B )  +  ( 1  x.  B ) )
2119mulid2i 7552 . . . . . . . 8  |-  ( 1  x.  B )  =  B
2221oveq2i 5677 . . . . . . 7  |-  ( ( 9  x.  B )  +  ( 1  x.  B ) )  =  ( ( 9  x.  B )  +  B
)
2317, 20, 223eqtri 2113 . . . . . 6  |-  (; 1 0  x.  B
)  =  ( ( 9  x.  B )  +  B )
2414, 23oveq12i 5678 . . . . 5  |-  ( ( (; 1 0 ^ 2 )  x.  A )  +  (; 1 0  x.  B
) )  =  ( ( (; 9 9  x.  A
)  +  A )  +  ( ( 9  x.  B )  +  B ) )
2524oveq1i 5676 . . . 4  |-  ( ( ( (; 1 0 ^ 2 )  x.  A )  +  (; 1 0  x.  B
) )  +  C
)  =  ( ( ( (; 9 9  x.  A
)  +  A )  +  ( ( 9  x.  B )  +  B ) )  +  C )
268, 10mulcli 7554 . . . . . 6  |-  (; 9 9  x.  A
)  e.  CC
2718, 19mulcli 7554 . . . . . 6  |-  ( 9  x.  B )  e.  CC
28 add4 7704 . . . . . . 7  |-  ( ( ( (; 9 9  x.  A
)  e.  CC  /\  A  e.  CC )  /\  ( ( 9  x.  B )  e.  CC  /\  B  e.  CC ) )  ->  ( (
(; 9 9  x.  A
)  +  A )  +  ( ( 9  x.  B )  +  B ) )  =  ( ( (; 9 9  x.  A
)  +  ( 9  x.  B ) )  +  ( A  +  B ) ) )
2928oveq1d 5681 . . . . . 6  |-  ( ( ( (; 9 9  x.  A
)  e.  CC  /\  A  e.  CC )  /\  ( ( 9  x.  B )  e.  CC  /\  B  e.  CC ) )  ->  ( (
( (; 9 9  x.  A
)  +  A )  +  ( ( 9  x.  B )  +  B ) )  +  C )  =  ( ( ( (; 9 9  x.  A
)  +  ( 9  x.  B ) )  +  ( A  +  B ) )  +  C ) )
3026, 10, 27, 19, 29mp4an 419 . . . . 5  |-  ( ( ( (; 9 9  x.  A
)  +  A )  +  ( ( 9  x.  B )  +  B ) )  +  C )  =  ( ( ( (; 9 9  x.  A
)  +  ( 9  x.  B ) )  +  ( A  +  B ) )  +  C )
3126, 27addcli 7553 . . . . . 6  |-  ( (; 9
9  x.  A )  +  ( 9  x.  B ) )  e.  CC
3210, 19addcli 7553 . . . . . 6  |-  ( A  +  B )  e.  CC
33 3dvds2dec.c . . . . . . 7  |-  C  e. 
NN0
3433nn0cni 8746 . . . . . 6  |-  C  e.  CC
3531, 32, 34addassi 7557 . . . . 5  |-  ( ( ( (; 9 9  x.  A
)  +  ( 9  x.  B ) )  +  ( A  +  B ) )  +  C )  =  ( ( (; 9 9  x.  A
)  +  ( 9  x.  B ) )  +  ( ( A  +  B )  +  C ) )
36 9t11e99 9067 . . . . . . . . . . 11  |-  ( 9  x. ; 1 1 )  = ; 9
9
3736eqcomi 2093 . . . . . . . . . 10  |- ; 9 9  =  ( 9  x. ; 1 1 )
3837oveq1i 5676 . . . . . . . . 9  |-  (; 9 9  x.  A
)  =  ( ( 9  x. ; 1 1 )  x.  A )
39 1nn0 8750 . . . . . . . . . . . 12  |-  1  e.  NN0
4039, 39deccl 8952 . . . . . . . . . . 11  |- ; 1 1  e.  NN0
4140nn0cni 8746 . . . . . . . . . 10  |- ; 1 1  e.  CC
4218, 41, 10mulassi 7558 . . . . . . . . 9  |-  ( ( 9  x. ; 1 1 )  x.  A )  =  ( 9  x.  (; 1 1  x.  A
) )
4338, 42eqtri 2109 . . . . . . . 8  |-  (; 9 9  x.  A
)  =  ( 9  x.  (; 1 1  x.  A
) )
4443oveq1i 5676 . . . . . . 7  |-  ( (; 9
9  x.  A )  +  ( 9  x.  B ) )  =  ( ( 9  x.  (; 1 1  x.  A
) )  +  ( 9  x.  B ) )
4541, 10mulcli 7554 . . . . . . . . 9  |-  (; 1 1  x.  A
)  e.  CC
4618, 45, 19adddii 7559 . . . . . . . 8  |-  ( 9  x.  ( (; 1 1  x.  A
)  +  B ) )  =  ( ( 9  x.  (; 1 1  x.  A
) )  +  ( 9  x.  B ) )
4746eqcomi 2093 . . . . . . 7  |-  ( ( 9  x.  (; 1 1  x.  A
) )  +  ( 9  x.  B ) )  =  ( 9  x.  ( (; 1 1  x.  A
)  +  B ) )
48 3t3e9 8634 . . . . . . . . . 10  |-  ( 3  x.  3 )  =  9
4948eqcomi 2093 . . . . . . . . 9  |-  9  =  ( 3  x.  3 )
5049oveq1i 5676 . . . . . . . 8  |-  ( 9  x.  ( (; 1 1  x.  A
)  +  B ) )  =  ( ( 3  x.  3 )  x.  ( (; 1 1  x.  A
)  +  B ) )
51 3cn 8558 . . . . . . . . 9  |-  3  e.  CC
5245, 19addcli 7553 . . . . . . . . 9  |-  ( (; 1
1  x.  A )  +  B )  e.  CC
5351, 51, 52mulassi 7558 . . . . . . . 8  |-  ( ( 3  x.  3 )  x.  ( (; 1 1  x.  A
)  +  B ) )  =  ( 3  x.  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) ) )
5450, 53eqtri 2109 . . . . . . 7  |-  ( 9  x.  ( (; 1 1  x.  A
)  +  B ) )  =  ( 3  x.  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) ) )
5544, 47, 543eqtri 2113 . . . . . 6  |-  ( (; 9
9  x.  A )  +  ( 9  x.  B ) )  =  ( 3  x.  (
3  x.  ( (; 1
1  x.  A )  +  B ) ) )
5655oveq1i 5676 . . . . 5  |-  ( ( (; 9 9  x.  A
)  +  ( 9  x.  B ) )  +  ( ( A  +  B )  +  C ) )  =  ( ( 3  x.  ( 3  x.  (
(; 1 1  x.  A
)  +  B ) ) )  +  ( ( A  +  B
)  +  C ) )
5730, 35, 563eqtri 2113 . . . 4  |-  ( ( ( (; 9 9  x.  A
)  +  A )  +  ( ( 9  x.  B )  +  B ) )  +  C )  =  ( ( 3  x.  (
3  x.  ( (; 1
1  x.  A )  +  B ) ) )  +  ( ( A  +  B )  +  C ) )
583, 25, 573eqtri 2113 . . 3  |- ;; A B C  =  (
( 3  x.  (
3  x.  ( (; 1
1  x.  A )  +  B ) ) )  +  ( ( A  +  B )  +  C ) )
5958breq2i 3859 . 2  |-  ( 3 
|| ;; A B C  <->  3  ||  ( ( 3  x.  ( 3  x.  (
(; 1 1  x.  A
)  +  B ) ) )  +  ( ( A  +  B
)  +  C ) ) )
60 3z 8840 . . 3  |-  3  e.  ZZ
611nn0zi 8833 . . . . 5  |-  A  e.  ZZ
622nn0zi 8833 . . . . 5  |-  B  e.  ZZ
63 zaddcl 8851 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  +  B
)  e.  ZZ )
6461, 62, 63mp2an 418 . . . 4  |-  ( A  +  B )  e.  ZZ
6533nn0zi 8833 . . . 4  |-  C  e.  ZZ
66 zaddcl 8851 . . . 4  |-  ( ( ( A  +  B
)  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( A  +  B )  +  C
)  e.  ZZ )
6764, 65, 66mp2an 418 . . 3  |-  ( ( A  +  B )  +  C )  e.  ZZ
6840nn0zi 8833 . . . . . . . 8  |- ; 1 1  e.  ZZ
69 zmulcl 8864 . . . . . . . 8  |-  ( (; 1
1  e.  ZZ  /\  A  e.  ZZ )  ->  (; 1 1  x.  A
)  e.  ZZ )
7068, 61, 69mp2an 418 . . . . . . 7  |-  (; 1 1  x.  A
)  e.  ZZ
71 zaddcl 8851 . . . . . . 7  |-  ( ( (; 1 1  x.  A
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( (; 1 1  x.  A
)  +  B )  e.  ZZ )
7270, 62, 71mp2an 418 . . . . . 6  |-  ( (; 1
1  x.  A )  +  B )  e.  ZZ
73 zmulcl 8864 . . . . . 6  |-  ( ( 3  e.  ZZ  /\  ( (; 1 1  x.  A
)  +  B )  e.  ZZ )  -> 
( 3  x.  (
(; 1 1  x.  A
)  +  B ) )  e.  ZZ )
7460, 72, 73mp2an 418 . . . . 5  |-  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) )  e.  ZZ
75 zmulcl 8864 . . . . 5  |-  ( ( 3  e.  ZZ  /\  ( 3  x.  (
(; 1 1  x.  A
)  +  B ) )  e.  ZZ )  ->  ( 3  x.  ( 3  x.  (
(; 1 1  x.  A
)  +  B ) ) )  e.  ZZ )
7660, 74, 75mp2an 418 . . . 4  |-  ( 3  x.  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) ) )  e.  ZZ
77 dvdsmul1 11157 . . . . 5  |-  ( ( 3  e.  ZZ  /\  ( 3  x.  (
(; 1 1  x.  A
)  +  B ) )  e.  ZZ )  ->  3  ||  (
3  x.  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) ) ) )
7860, 74, 77mp2an 418 . . . 4  |-  3  ||  ( 3  x.  (
3  x.  ( (; 1
1  x.  A )  +  B ) ) )
7976, 78pm3.2i 267 . . 3  |-  ( ( 3  x.  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) ) )  e.  ZZ  /\  3  ||  ( 3  x.  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) ) ) )
80 dvdsadd2b 11182 . . 3  |-  ( ( 3  e.  ZZ  /\  ( ( A  +  B )  +  C
)  e.  ZZ  /\  ( ( 3  x.  ( 3  x.  (
(; 1 1  x.  A
)  +  B ) ) )  e.  ZZ  /\  3  ||  ( 3  x.  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) ) ) ) )  ->  ( 3  ||  ( ( A  +  B )  +  C
)  <->  3  ||  (
( 3  x.  (
3  x.  ( (; 1
1  x.  A )  +  B ) ) )  +  ( ( A  +  B )  +  C ) ) ) )
8160, 67, 79, 80mp3an 1274 . 2  |-  ( 3 
||  ( ( A  +  B )  +  C )  <->  3  ||  ( ( 3  x.  ( 3  x.  (
(; 1 1  x.  A
)  +  B ) ) )  +  ( ( A  +  B
)  +  C ) ) )
8259, 81bitr4i 186 1  |-  ( 3 
|| ;; A B C  <->  3  ||  ( ( A  +  B )  +  C
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   class class class wbr 3851  (class class class)co 5666   CCcc 7409   0cc0 7411   1c1 7412    + caddc 7414    x. cmul 7416   2c2 8534   3c3 8535   9c9 8541   NN0cn0 8734   ZZcz 8811  ;cdc 8938   ^cexp 10015    || cdvds 11135
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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-3 8543  df-4 8544  df-5 8545  df-6 8546  df-7 8547  df-8 8548  df-9 8549  df-n0 8735  df-z 8812  df-dec 8939  df-uz 9081  df-iseq 9914  df-seq3 9915  df-exp 10016  df-dvds 11136
This theorem is referenced by: (None)
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