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Theorem dvdsadd2b 11865
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
StepHypRef Expression
1 simpl1 1002 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  e.  ZZ )
2 simpl3l 1054 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  C  e.  ZZ )
3 simpl2 1003 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  B  e.  ZZ )
4 simpl3r 1055 . . 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 11850 . . . 4  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  ||  C  /\  A  ||  B )  ->  A  ||  ( C  +  B )
) )
76imp 124 . . 3  |-  ( ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  ||  C  /\  A  ||  B ) )  ->  A  ||  ( C  +  B )
)
81, 2, 3, 4, 5, 7syl32anc 1257 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  ( C  +  B
) )
9 simpl1 1002 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  e.  ZZ )
10 simp3l 1027 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  C  e.  ZZ )
11 simp2 1000 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  B  e.  ZZ )
12 zaddcl 9311 . . . . . 6  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  +  B
)  e.  ZZ )
1310, 11, 12syl2anc 411 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  -> 
( C  +  B
)  e.  ZZ )
1413adantr 276 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  ( C  +  B )  e.  ZZ )
1510znegcld 9395 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  -u C  e.  ZZ )
1615adantr 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 1055 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  C )
19 simpl3l 1054 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  C  e.  ZZ )
20 dvdsnegb 11833 . . . . . 6  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  C  <->  A 
||  -u C ) )
219, 19, 20syl2anc 411 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  ( A  ||  C  <->  A  ||  -u C
) )
2218, 21mpbid 147 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  -u C )
23 dvds2add 11850 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  -u C  e.  ZZ )  ->  ( ( A 
||  ( C  +  B )  /\  A  ||  -u C )  ->  A  ||  ( ( C  +  B )  +  -u C ) ) )
2423imp 124 . . . 4  |-  ( ( ( A  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  -u C  e.  ZZ )  /\  ( A  ||  ( C  +  B
)  /\  A  ||  -u C
) )  ->  A  ||  ( ( C  +  B )  +  -u C ) )
259, 14, 16, 17, 22, 24syl32anc 1257 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  ( ( C  +  B )  +  -u C ) )
26 simpl2 1003 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  B  e.  ZZ )
2712ancoms 268 . . . . . . 7  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( C  +  B
)  e.  ZZ )
2827zcnd 9394 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( C  +  B
)  e.  CC )
29 zcn 9276 . . . . . . 7  |-  ( C  e.  ZZ  ->  C  e.  CC )
3029adantl 277 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  C  e.  CC )
3128, 30negsubd 8292 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  +  -u C )  =  ( ( C  +  B
)  -  C ) )
32 zcn 9276 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
3332adantr 276 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  B  e.  CC )
3430, 33pncan2d 8288 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  -  C
)  =  B )
3531, 34eqtrd 2222 . . . 4  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  +  -u C )  =  B )
3626, 19, 35syl2anc 411 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  (
( C  +  B
)  +  -u C
)  =  B )
3725, 36breqtrd 4044 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  B )
388, 37impbida 596 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  -> 
( A  ||  B  <->  A 
||  ( C  +  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5891   CCcc 7827    + caddc 7832    - cmin 8146   -ucneg 8147   ZZcz 9271    || cdvds 11812
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-0id 7937  ax-rnegex 7938  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-ltadd 7945
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-inn 8938  df-n0 9195  df-z 9272  df-dvds 11813
This theorem is referenced by:  dvdsaddre2b  11866  3dvdsdec  11888  3dvds2dec  11889  2sqlem3  14848
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