Theorem dvdsadd2b 12391

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 12376	. . . 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 9509	. . . . . 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 9594	. . . . 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 12359	. . . . . 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 12376	. . . . 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 9593	. . . . . 6 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( C + B ) e. CC )
29		zcn 9474	. . . . . . 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 8486	. . . . 5 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( ( C + B ) + -u C ) = ( ( C + B ) - C ) )
32		zcn 9474	. . . . . . 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 8482	. . . . 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 4112	. 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 4086  (class class class)co 6013   CCcc 8020   + caddc 8025   - cmin 8340   -ucneg 8341   ZZcz 9469   || cdvds 12338

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

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 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-dvds 12339

This theorem is referenced by:  dvdsaddre2b  12392  3dvdsdec  12416  3dvds2dec  12417  2sqlem3  15836