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

Step	Hyp	Ref	Expression
1		3dvdsdec.a	. . . . 5 |- A e. NN0
2		3dvdsdec.b	. . . . 5 |- B e. NN0
3	1, 2	3dec 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 )
5	4	oveq1i 5676	. . . . . . 7 |- ( (; 1 0 ^ 2 ) x. A ) = ( (; 9 9 + 1 ) x. A )
6		9nn0 8758	. . . . . . . . . 10 |- 9 e. NN0
7	6, 6	deccl 8952	. . . . . . . . 9 |- ; 9 9 e. NN0
8	7	nn0cni 8746	. . . . . . . 8 |- ; 9 9 e. CC
9		ax-1cn 7499	. . . . . . . 8 |- 1 e. CC
10	1	nn0cni 8746	. . . . . . . 8 |- A e. CC
11	8, 9, 10	adddiri 7560	. . . . . . 7 |- ( (; 9 9 + 1 ) x. A ) = ( (; 9 9 x. A ) + ( 1 x. A ) )
12	10	mulid2i 7552	. . . . . . . 8 |- ( 1 x. A ) = A
13	12	oveq2i 5677	. . . . . . 7 |- ( (; 9 9 x. A ) + ( 1 x. A ) ) = ( (; 9 9 x. A ) + A )
14	5, 11, 13	3eqtri 2113	. . . . . 6 |- ( (; 1 0 ^ 2 ) x. A ) = ( (; 9 9 x. A ) + A )
15		9p1e10 8940	. . . . . . . . 9 |- ( 9 + 1 ) = ; 1 0
16	15	eqcomi 2093	. . . . . . . 8 |- ; 1 0 = ( 9 + 1 )
17	16	oveq1i 5676	. . . . . . 7 |- (; 1 0 x. B ) = ( ( 9 + 1 ) x. B )
18		9cn 8571	. . . . . . . 8 |- 9 e. CC
19	2	nn0cni 8746	. . . . . . . 8 |- B e. CC
20	18, 9, 19	adddiri 7560	. . . . . . 7 |- ( ( 9 + 1 ) x. B ) = ( ( 9 x. B ) + ( 1 x. B ) )
21	19	mulid2i 7552	. . . . . . . 8 |- ( 1 x. B ) = B
22	21	oveq2i 5677	. . . . . . 7 |- ( ( 9 x. B ) + ( 1 x. B ) ) = ( ( 9 x. B ) + B )
23	17, 20, 22	3eqtri 2113	. . . . . 6 |- (; 1 0 x. B ) = ( ( 9 x. B ) + B )
24	14, 23	oveq12i 5678	. . . . 5 |- ( ( (; 1 0 ^ 2 ) x. A ) + (; 1 0 x. B ) ) = ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) )
25	24	oveq1i 5676	. . . 4 |- ( ( ( (; 1 0 ^ 2 ) x. A ) + (; 1 0 x. B ) ) + C ) = ( ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) + C )
26	8, 10	mulcli 7554	. . . . . 6 |- (; 9 9 x. A ) e. CC
27	18, 19	mulcli 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 ) ) )
29	28	oveq1d 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 ) )
30	26, 10, 27, 19, 29	mp4an 419	. . . . 5 |- ( ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) + C ) = ( ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( A + B ) ) + C )
31	26, 27	addcli 7553	. . . . . 6 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) e. CC
32	10, 19	addcli 7553	. . . . . 6 |- ( A + B ) e. CC
33		3dvds2dec.c	. . . . . . 7 |- C e. NN0
34	33	nn0cni 8746	. . . . . 6 |- C e. CC
35	31, 32, 34	addassi 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
37	36	eqcomi 2093	. . . . . . . . . 10 |- ; 9 9 = ( 9 x. ; 1 1 )
38	37	oveq1i 5676	. . . . . . . . 9 |- (; 9 9 x. A ) = ( ( 9 x. ; 1 1 ) x. A )
39		1nn0 8750	. . . . . . . . . . . 12 |- 1 e. NN0
40	39, 39	deccl 8952	. . . . . . . . . . 11 |- ; 1 1 e. NN0
41	40	nn0cni 8746	. . . . . . . . . 10 |- ; 1 1 e. CC
42	18, 41, 10	mulassi 7558	. . . . . . . . 9 |- ( ( 9 x. ; 1 1 ) x. A ) = ( 9 x. (; 1 1 x. A ) )
43	38, 42	eqtri 2109	. . . . . . . 8 |- (; 9 9 x. A ) = ( 9 x. (; 1 1 x. A ) )
44	43	oveq1i 5676	. . . . . . 7 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) = ( ( 9 x. (; 1 1 x. A ) ) + ( 9 x. B ) )
45	41, 10	mulcli 7554	. . . . . . . . 9 |- (; 1 1 x. A ) e. CC
46	18, 45, 19	adddii 7559	. . . . . . . 8 |- ( 9 x. ( (; 1 1 x. A ) + B ) ) = ( ( 9 x. (; 1 1 x. A ) ) + ( 9 x. B ) )
47	46	eqcomi 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
49	48	eqcomi 2093	. . . . . . . . 9 |- 9 = ( 3 x. 3 )
50	49	oveq1i 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
52	45, 19	addcli 7553	. . . . . . . . 9 |- ( (; 1 1 x. A ) + B ) e. CC
53	51, 51, 52	mulassi 7558	. . . . . . . 8 |- ( ( 3 x. 3 ) x. ( (; 1 1 x. A ) + B ) ) = ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
54	50, 53	eqtri 2109	. . . . . . 7 |- ( 9 x. ( (; 1 1 x. A ) + B ) ) = ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
55	44, 47, 54	3eqtri 2113	. . . . . 6 |- ( (; 9 9 x. A ) + ( 9 x. B ) ) = ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
56	55	oveq1i 5676	. . . . 5 |- ( ( (; 9 9 x. A ) + ( 9 x. B ) ) + ( ( A + B ) + C ) ) = ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) )
57	30, 35, 56	3eqtri 2113	. . . 4 |- ( ( ( (; 9 9 x. A ) + A ) + ( ( 9 x. B ) + B ) ) + C ) = ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) )
58	3, 25, 57	3eqtri 2113	. . 3 |- ;; A B C = ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) )
59	58	breq2i 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
61	1	nn0zi 8833	. . . . 5 |- A e. ZZ
62	2	nn0zi 8833	. . . . 5 |- B e. ZZ
63		zaddcl 8851	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + B ) e. ZZ )
64	61, 62, 63	mp2an 418	. . . 4 |- ( A + B ) e. ZZ
65	33	nn0zi 8833	. . . 4 |- C e. ZZ
66		zaddcl 8851	. . . 4 |- ( ( ( A + B ) e. ZZ /\ C e. ZZ ) -> ( ( A + B ) + C ) e. ZZ )
67	64, 65, 66	mp2an 418	. . 3 |- ( ( A + B ) + C ) e. ZZ
68	40	nn0zi 8833	. . . . . . . 8 |- ; 1 1 e. ZZ
69		zmulcl 8864	. . . . . . . 8 |- ( (; 1 1 e. ZZ /\ A e. ZZ ) -> (; 1 1 x. A ) e. ZZ )
70	68, 61, 69	mp2an 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 )
72	70, 62, 71	mp2an 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 )
74	60, 72, 73	mp2an 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 )
76	60, 74, 75	mp2an 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 ) ) ) )
78	60, 74, 77	mp2an 418	. . . 4 |- 3 || ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) )
79	76, 78	pm3.2i 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 ) ) ) )
81	60, 67, 79, 80	mp3an 1274	. 2 |- ( 3 || ( ( A + B ) + C ) <-> 3 || ( ( 3 x. ( 3 x. ( (; 1 1 x. A ) + B ) ) ) + ( ( A + B ) + C ) ) )
82	59, 81	bitr4i 186	1 |- ( 3 || ;; A B C <-> 3 || ( ( A + B ) + C ) )

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)