Theorem dvdsadd2b 12526

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 1027	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> A e. ZZ )
2		simpl3l 1079	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> C e. ZZ )
3		simpl2 1028	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> B e. ZZ )
4		simpl3r 1080	. . 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 12511	. . . 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 1282	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> A || ( C + B ) )
9		simpl1 1027	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A e. ZZ )
10		simp3l 1052	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> C e. ZZ )
11		simp2 1025	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> B e. ZZ )
12		zaddcl 9617	. . . . . 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 9702	. . . . 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 1080	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || C )
19		simpl3l 1079	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> C e. ZZ )
20		dvdsnegb 12494	. . . . . 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 12511	. . . . 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 1282	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || ( ( C + B ) + -u C ) )
26		simpl2 1028	. . . 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 9701	. . . . . 6 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( C + B ) e. CC )
29		zcn 9582	. . . . . . 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 8590	. . . . 5 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( ( C + B ) + -u C ) = ( ( C + B ) - C ) )
32		zcn 9582	. . . . . . 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 8586	. . . . 5 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( ( C + B ) - C ) = B )
35	31, 34	eqtrd 2265	. . . 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 4135	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || B )
38	8, 37	impbida 600	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 1005   = wceq 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   CCcc 8125   + caddc 8130   - cmin 8444   -ucneg 8445   ZZcz 9577   || cdvds 12473

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-dvds 12474

This theorem is referenced by:  dvdsaddre2b  12527  3dvdsdec  12551  3dvds2dec  12552  2sqlem3  15990  eupth2lem3lem3fi  16465