Theorem dvdsadd2b 12346

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 1024	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> A e. ZZ )
2		simpl3l 1076	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> C e. ZZ )
3		simpl2 1025	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> B e. ZZ )
4		simpl3r 1077	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> A || C )
5		simpr 110	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> A || B )
6		dvds2add 12331	. . . 4 |- ( ( A e. ZZ /\ C e. ZZ /\ B e. ZZ ) -> ( ( A || C /\ A || B ) -> A || ( C + B ) ) )
7	6	imp 124	. . 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 1279	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> A || ( C + B ) )
9		simpl1 1024	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A e. ZZ )
10		simp3l 1049	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> C e. ZZ )
11		simp2 1022	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> B e. ZZ )
12		zaddcl 9482	. . . . . 6 |- ( ( C e. ZZ /\ B e. ZZ ) -> ( C + B ) e. ZZ )
13	10, 11, 12	syl2anc 411	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( C + B ) e. ZZ )
14	13	adantr 276	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( C + B ) e. ZZ )
15	10	znegcld 9567	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> -u C e. ZZ )
16	15	adantr 276	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> -u C e. ZZ )
17		simpr 110	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || ( C + B ) )
18		simpl3r 1077	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || C )
19		simpl3l 1076	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> C e. ZZ )
20		dvdsnegb 12314	. . . . . 6 |- ( ( A e. ZZ /\ C e. ZZ ) -> ( A || C <-> A || -u C ) )
21	9, 19, 20	syl2anc 411	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( A || C <-> A || -u C ) )
22	18, 21	mpbid 147	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || -u C )
23		dvds2add 12331	. . . . 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 124	. . . 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 1279	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || ( ( C + B ) + -u C ) )
26		simpl2 1025	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> B e. ZZ )
27	12	ancoms 268	. . . . . . 7 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( C + B ) e. ZZ )
28	27	zcnd 9566	. . . . . 6 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( C + B ) e. CC )
29		zcn 9447	. . . . . . 7 |- ( C e. ZZ -> C e. CC )
30	29	adantl 277	. . . . . 6 |- ( ( B e. ZZ /\ C e. ZZ ) -> C e. CC )
31	28, 30	negsubd 8459	. . . . 5 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( ( C + B ) + -u C ) = ( ( C + B ) - C ) )
32		zcn 9447	. . . . . . 7 |- ( B e. ZZ -> B e. CC )
33	32	adantr 276	. . . . . 6 |- ( ( B e. ZZ /\ C e. ZZ ) -> B e. CC )
34	30, 33	pncan2d 8455	. . . . 5 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( ( C + B ) - C ) = B )
35	31, 34	eqtrd 2262	. . . 4 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( ( C + B ) + -u C ) = B )
36	26, 19, 35	syl2anc 411	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) + -u C ) = B )
37	25, 36	breqtrd 4108	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || B )
38	8, 37	impbida 598	1 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> ( A || B <-> A || ( C + B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4082  (class class class)co 6000   CCcc 7993   + caddc 7998   - cmin 8313   -ucneg 8314   ZZcz 9442   || cdvds 12293

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-dvds 12294

This theorem is referenced by:  dvdsaddre2b  12347  3dvdsdec  12371  3dvds2dec  12372  2sqlem3  15790