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Theorem 3dvds2dec 11903
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 10726 . . . 4  |- ;; A B C  =  (
( ( (; 1 0 ^ 2 )  x.  A )  +  (; 1 0  x.  B
) )  +  C
)
4 sq10e99m1 10725 . . . . . . . 8  |-  (; 1 0 ^ 2 )  =  (; 9 9  +  1 )
54oveq1i 5906 . . . . . . 7  |-  ( (; 1
0 ^ 2 )  x.  A )  =  ( (; 9 9  +  1 )  x.  A )
6 9nn0 9230 . . . . . . . . . 10  |-  9  e.  NN0
76, 6deccl 9428 . . . . . . . . 9  |- ; 9 9  e.  NN0
87nn0cni 9218 . . . . . . . 8  |- ; 9 9  e.  CC
9 ax-1cn 7934 . . . . . . . 8  |-  1  e.  CC
101nn0cni 9218 . . . . . . . 8  |-  A  e.  CC
118, 9, 10adddiri 7998 . . . . . . 7  |-  ( (; 9
9  +  1 )  x.  A )  =  ( (; 9 9  x.  A
)  +  ( 1  x.  A ) )
1210mullidi 7990 . . . . . . . 8  |-  ( 1  x.  A )  =  A
1312oveq2i 5907 . . . . . . 7  |-  ( (; 9
9  x.  A )  +  ( 1  x.  A ) )  =  ( (; 9 9  x.  A
)  +  A )
145, 11, 133eqtri 2214 . . . . . 6  |-  ( (; 1
0 ^ 2 )  x.  A )  =  ( (; 9 9  x.  A
)  +  A )
15 9p1e10 9416 . . . . . . . . 9  |-  ( 9  +  1 )  = ; 1
0
1615eqcomi 2193 . . . . . . . 8  |- ; 1 0  =  ( 9  +  1 )
1716oveq1i 5906 . . . . . . 7  |-  (; 1 0  x.  B
)  =  ( ( 9  +  1 )  x.  B )
18 9cn 9037 . . . . . . . 8  |-  9  e.  CC
192nn0cni 9218 . . . . . . . 8  |-  B  e.  CC
2018, 9, 19adddiri 7998 . . . . . . 7  |-  ( ( 9  +  1 )  x.  B )  =  ( ( 9  x.  B )  +  ( 1  x.  B ) )
2119mullidi 7990 . . . . . . . 8  |-  ( 1  x.  B )  =  B
2221oveq2i 5907 . . . . . . 7  |-  ( ( 9  x.  B )  +  ( 1  x.  B ) )  =  ( ( 9  x.  B )  +  B
)
2317, 20, 223eqtri 2214 . . . . . 6  |-  (; 1 0  x.  B
)  =  ( ( 9  x.  B )  +  B )
2414, 23oveq12i 5908 . . . . 5  |-  ( ( (; 1 0 ^ 2 )  x.  A )  +  (; 1 0  x.  B
) )  =  ( ( (; 9 9  x.  A
)  +  A )  +  ( ( 9  x.  B )  +  B ) )
2524oveq1i 5906 . . . 4  |-  ( ( ( (; 1 0 ^ 2 )  x.  A )  +  (; 1 0  x.  B
) )  +  C
)  =  ( ( ( (; 9 9  x.  A
)  +  A )  +  ( ( 9  x.  B )  +  B ) )  +  C )
268, 10mulcli 7992 . . . . . 6  |-  (; 9 9  x.  A
)  e.  CC
2718, 19mulcli 7992 . . . . . 6  |-  ( 9  x.  B )  e.  CC
28 add4 8148 . . . . . . 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 5911 . . . . . 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 427 . . . . 5  |-  ( ( ( (; 9 9  x.  A
)  +  A )  +  ( ( 9  x.  B )  +  B ) )  +  C )  =  ( ( ( (; 9 9  x.  A
)  +  ( 9  x.  B ) )  +  ( A  +  B ) )  +  C )
3126, 27addcli 7991 . . . . . 6  |-  ( (; 9
9  x.  A )  +  ( 9  x.  B ) )  e.  CC
3210, 19addcli 7991 . . . . . 6  |-  ( A  +  B )  e.  CC
33 3dvds2dec.c . . . . . . 7  |-  C  e. 
NN0
3433nn0cni 9218 . . . . . 6  |-  C  e.  CC
3531, 32, 34addassi 7995 . . . . 5  |-  ( ( ( (; 9 9  x.  A
)  +  ( 9  x.  B ) )  +  ( A  +  B ) )  +  C )  =  ( ( (; 9 9  x.  A
)  +  ( 9  x.  B ) )  +  ( ( A  +  B )  +  C ) )
36 9t11e99 9543 . . . . . . . . . . 11  |-  ( 9  x. ; 1 1 )  = ; 9
9
3736eqcomi 2193 . . . . . . . . . 10  |- ; 9 9  =  ( 9  x. ; 1 1 )
3837oveq1i 5906 . . . . . . . . 9  |-  (; 9 9  x.  A
)  =  ( ( 9  x. ; 1 1 )  x.  A )
39 1nn0 9222 . . . . . . . . . . . 12  |-  1  e.  NN0
4039, 39deccl 9428 . . . . . . . . . . 11  |- ; 1 1  e.  NN0
4140nn0cni 9218 . . . . . . . . . 10  |- ; 1 1  e.  CC
4218, 41, 10mulassi 7996 . . . . . . . . 9  |-  ( ( 9  x. ; 1 1 )  x.  A )  =  ( 9  x.  (; 1 1  x.  A
) )
4338, 42eqtri 2210 . . . . . . . 8  |-  (; 9 9  x.  A
)  =  ( 9  x.  (; 1 1  x.  A
) )
4443oveq1i 5906 . . . . . . 7  |-  ( (; 9
9  x.  A )  +  ( 9  x.  B ) )  =  ( ( 9  x.  (; 1 1  x.  A
) )  +  ( 9  x.  B ) )
4541, 10mulcli 7992 . . . . . . . . 9  |-  (; 1 1  x.  A
)  e.  CC
4618, 45, 19adddii 7997 . . . . . . . 8  |-  ( 9  x.  ( (; 1 1  x.  A
)  +  B ) )  =  ( ( 9  x.  (; 1 1  x.  A
) )  +  ( 9  x.  B ) )
4746eqcomi 2193 . . . . . . 7  |-  ( ( 9  x.  (; 1 1  x.  A
) )  +  ( 9  x.  B ) )  =  ( 9  x.  ( (; 1 1  x.  A
)  +  B ) )
48 3t3e9 9106 . . . . . . . . . 10  |-  ( 3  x.  3 )  =  9
4948eqcomi 2193 . . . . . . . . 9  |-  9  =  ( 3  x.  3 )
5049oveq1i 5906 . . . . . . . 8  |-  ( 9  x.  ( (; 1 1  x.  A
)  +  B ) )  =  ( ( 3  x.  3 )  x.  ( (; 1 1  x.  A
)  +  B ) )
51 3cn 9024 . . . . . . . . 9  |-  3  e.  CC
5245, 19addcli 7991 . . . . . . . . 9  |-  ( (; 1
1  x.  A )  +  B )  e.  CC
5351, 51, 52mulassi 7996 . . . . . . . 8  |-  ( ( 3  x.  3 )  x.  ( (; 1 1  x.  A
)  +  B ) )  =  ( 3  x.  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) ) )
5450, 53eqtri 2210 . . . . . . 7  |-  ( 9  x.  ( (; 1 1  x.  A
)  +  B ) )  =  ( 3  x.  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) ) )
5544, 47, 543eqtri 2214 . . . . . 6  |-  ( (; 9
9  x.  A )  +  ( 9  x.  B ) )  =  ( 3  x.  (
3  x.  ( (; 1
1  x.  A )  +  B ) ) )
5655oveq1i 5906 . . . . 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 2214 . . . 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 2214 . . 3  |- ;; A B C  =  (
( 3  x.  (
3  x.  ( (; 1
1  x.  A )  +  B ) ) )  +  ( ( A  +  B )  +  C ) )
5958breq2i 4026 . 2  |-  ( 3 
|| ;; A B C  <->  3  ||  ( ( 3  x.  ( 3  x.  (
(; 1 1  x.  A
)  +  B ) ) )  +  ( ( A  +  B
)  +  C ) ) )
60 3z 9312 . . 3  |-  3  e.  ZZ
611nn0zi 9305 . . . . 5  |-  A  e.  ZZ
622nn0zi 9305 . . . . 5  |-  B  e.  ZZ
63 zaddcl 9323 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  +  B
)  e.  ZZ )
6461, 62, 63mp2an 426 . . . 4  |-  ( A  +  B )  e.  ZZ
6533nn0zi 9305 . . . 4  |-  C  e.  ZZ
66 zaddcl 9323 . . . 4  |-  ( ( ( A  +  B
)  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( A  +  B )  +  C
)  e.  ZZ )
6764, 65, 66mp2an 426 . . 3  |-  ( ( A  +  B )  +  C )  e.  ZZ
6840nn0zi 9305 . . . . . . . 8  |- ; 1 1  e.  ZZ
69 zmulcl 9336 . . . . . . . 8  |-  ( (; 1
1  e.  ZZ  /\  A  e.  ZZ )  ->  (; 1 1  x.  A
)  e.  ZZ )
7068, 61, 69mp2an 426 . . . . . . 7  |-  (; 1 1  x.  A
)  e.  ZZ
71 zaddcl 9323 . . . . . . 7  |-  ( ( (; 1 1  x.  A
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( (; 1 1  x.  A
)  +  B )  e.  ZZ )
7270, 62, 71mp2an 426 . . . . . 6  |-  ( (; 1
1  x.  A )  +  B )  e.  ZZ
73 zmulcl 9336 . . . . . 6  |-  ( ( 3  e.  ZZ  /\  ( (; 1 1  x.  A
)  +  B )  e.  ZZ )  -> 
( 3  x.  (
(; 1 1  x.  A
)  +  B ) )  e.  ZZ )
7460, 72, 73mp2an 426 . . . . 5  |-  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) )  e.  ZZ
75 zmulcl 9336 . . . . 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 426 . . . 4  |-  ( 3  x.  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) ) )  e.  ZZ
77 dvdsmul1 11852 . . . . 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 426 . . . 4  |-  3  ||  ( 3  x.  (
3  x.  ( (; 1
1  x.  A )  +  B ) ) )
7976, 78pm3.2i 272 . . 3  |-  ( ( 3  x.  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) ) )  e.  ZZ  /\  3  ||  ( 3  x.  ( 3  x.  ( (; 1 1  x.  A
)  +  B ) ) ) )
80 dvdsadd2b 11879 . . 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 1348 . 2  |-  ( 3 
||  ( ( A  +  B )  +  C )  <->  3  ||  ( ( 3  x.  ( 3  x.  (
(; 1 1  x.  A
)  +  B ) ) )  +  ( ( A  +  B
)  +  C ) ) )
8259, 81bitr4i 187 1  |-  ( 3 
|| ;; A B C  <->  3  ||  ( ( A  +  B )  +  C
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5896   CCcc 7839   0cc0 7841   1c1 7842    + caddc 7844    x. cmul 7846   2c2 9000   3c3 9001   9c9 9007   NN0cn0 9206   ZZcz 9283  ;cdc 9414   ^cexp 10550    || cdvds 11826
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-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959
This theorem depends on definitions:  df-bi 117  df-dc 836  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-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-5 9011  df-6 9012  df-7 9013  df-8 9014  df-9 9015  df-n0 9207  df-z 9284  df-dec 9415  df-uz 9559  df-seqfrec 10477  df-exp 10551  df-dvds 11827
This theorem is referenced by: (None)
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