Theorem dvdsadd2b 10450

Description: Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.)

Assertion

Ref	Expression
dvdsadd2b	|- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) )

Proof of Theorem dvdsadd2b

Step	Hyp	Ref	Expression
1		simpl1 942	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> A e. ZZ )
2		simpl3l 994	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> C e. ZZ )
3		simpl2 943	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> B e. ZZ )
4		simpl3r 995	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> A || C )
5		simpr 108	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> A || B )
6		dvds2add 10437	. . . 4 |- ( ( A e. ZZ /\ C e. ZZ /\ B e. ZZ ) -> ( ( A || C /\ A || B ) -> A || ( C + B ) ) )
7	6	imp 122	. . 3 |- ( ( ( A e. ZZ /\ C e. ZZ /\ B e. ZZ ) /\ ( A || C /\ A || B ) ) -> A || ( C + B ) )
8	1, 2, 3, 4, 5, 7	syl32anc 1178	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> A || ( C + B ) )
9		simpl1 942	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A e. ZZ )
10		simp3l 967	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> C e. ZZ )
11		simp2 940	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> B e. ZZ )
12		zaddcl 8524	. . . . . 6 |- ( ( C e. ZZ /\ B e. ZZ ) -> ( C + B ) e. ZZ )
13	10, 11, 12	syl2anc 403	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( C + B ) e. ZZ )
14	13	adantr 270	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( C + B ) e. ZZ )
15	10	znegcld 8604	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> -u C e. ZZ )
16	15	adantr 270	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> -u C e. ZZ )
17		simpr 108	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || ( C + B ) )
18		simpl3r 995	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || C )
19		simpl3l 994	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> C e. ZZ )
20		dvdsnegb 10420	. . . . . 6 |- ( ( A e. ZZ /\ C e. ZZ ) -> ( A || C <-> A || -u C ) )
21	9, 19, 20	syl2anc 403	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( A || C <-> A || -u C ) )
22	18, 21	mpbid 145	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || -u C )
23		dvds2add 10437	. . . . 5 |- ( ( A e. ZZ /\ ( C + B ) e. ZZ /\ -u C e. ZZ ) -> ( ( A || ( C + B ) /\ A || -u C ) -> A || ( ( C + B ) + -u C ) ) )
24	23	imp 122	. . . 4 |- ( ( ( A e. ZZ /\ ( C + B ) e. ZZ /\ -u C e. ZZ ) /\ ( A || ( C + B ) /\ A || -u C ) ) -> A || ( ( C + B ) + -u C ) )
25	9, 14, 16, 17, 22, 24	syl32anc 1178	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || ( ( C + B ) + -u C ) )
26		simpl2 943	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> B e. ZZ )
27	12	ancoms 264	. . . . . . 7 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( C + B ) e. ZZ )
28	27	zcnd 8603	. . . . . 6 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( C + B ) e. CC )
29		zcn 8489	. . . . . . 7 |- ( C e. ZZ -> C e. CC )
30	29	adantl 271	. . . . . 6 |- ( ( B e. ZZ /\ C e. ZZ ) -> C e. CC )
31	28, 30	negsubd 7544	. . . . 5 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( ( C + B ) + -u C ) = ( ( C + B ) - C ) )
32		zcn 8489	. . . . . . 7 |- ( B e. ZZ -> B e. CC )
33	32	adantr 270	. . . . . 6 |- ( ( B e. ZZ /\ C e. ZZ ) -> B e. CC )
34	30, 33	pncan2d 7540	. . . . 5 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( ( C + B ) - C ) = B )
35	31, 34	eqtrd 2115	. . . 4 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( ( C + B ) + -u C ) = B )
36	26, 19, 35	syl2anc 403	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) + -u C ) = B )
37	25, 36	breqtrd 3829	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || B )
38	8, 37	impbida 561	1 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 920   = wceq 1285   e. wcel 1434   class class class wbr 3805  (class class class)co 5563   CCcc 7093   + caddc 7098   - cmin 7398   -ucneg 7399   ZZcz 8484   || cdvds 10403

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-dvds 10404

This theorem is referenced by:  3dvdsdec  10472  3dvds2dec  10473