Theorem dvdsadd2b 12585

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 1031	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> A e. ZZ )
2		simpl3l 1083	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> C e. ZZ )
3		simpl2 1032	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> B e. ZZ )
4		simpl3r 1084	. . 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 12570	. . . 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 1286	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || B ) -> A || ( C + B ) )
9		simpl1 1031	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A e. ZZ )
10		simp3l 1056	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> C e. ZZ )
11		simp2 1029	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) -> B e. ZZ )
12		zaddcl 9663	. . . . . 6 |- ( ( C e. ZZ /\ B e. ZZ ) -> ( C + B ) e. ZZ )
13	10, 11, 12	syl2anc 415	. . . . 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 9749	. . . . 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 1084	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || C )
19		simpl3l 1083	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> C e. ZZ )
20		dvdsnegb 12553	. . . . . 6 |- ( ( A e. ZZ /\ C e. ZZ ) -> ( A || C <-> A || -u C ) )
21	9, 19, 20	syl2anc 415	. . . . 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 12570	. . . . 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 1286	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || ( ( C + B ) + -u C ) )
26		simpl2 1032	. . . 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 9748	. . . . . 6 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( C + B ) e. CC )
29		zcn 9628	. . . . . . 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 8633	. . . . 5 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( ( C + B ) + -u C ) = ( ( C + B ) - C ) )
32		zcn 9628	. . . . . . 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 8629	. . . . 5 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( ( C + B ) - C ) = B )
35	31, 34	eqtrd 2271	. . . 4 |- ( ( B e. ZZ /\ C e. ZZ ) -> ( ( C + B ) + -u C ) = B )
36	26, 19, 35	syl2anc 415	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> ( ( C + B ) + -u C ) = B )
37	25, 36	breqtrd 4151	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( C e. ZZ /\ A || C ) ) /\ A || ( C + B ) ) -> A || B )
38	8, 37	impbida 604	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 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   CCcc 8167   + caddc 8172   - cmin 8487   -ucneg 8488   ZZcz 9623   || cdvds 12532

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-dvds 12533

This theorem is referenced by:  dvdsaddre2b  12586  3dvdsdec  12610  3dvds2dec  12611  2sqlem3  16150  eupth2lem3lem3fi  16625