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Theorem dvdsadd2b 10450
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 942 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  e.  ZZ )
2 simpl3l 994 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  C  e.  ZZ )
3 simpl2 943 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  B  e.  ZZ )
4 simpl3r 995 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  C )
5 simpr 108 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  B )
6 dvds2add 10437 . . . 4  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  ||  C  /\  A  ||  B )  ->  A  ||  ( C  +  B )
) )
76imp 122 . . 3  |-  ( ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  ||  C  /\  A  ||  B ) )  ->  A  ||  ( C  +  B )
)
81, 2, 3, 4, 5, 7syl32anc 1178 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  ( C  +  B
) )
9 simpl1 942 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  e.  ZZ )
10 simp3l 967 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  C  e.  ZZ )
11 simp2 940 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  B  e.  ZZ )
12 zaddcl 8524 . . . . . 6  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  +  B
)  e.  ZZ )
1310, 11, 12syl2anc 403 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  -> 
( C  +  B
)  e.  ZZ )
1413adantr 270 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  ( C  +  B )  e.  ZZ )
1510znegcld 8604 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  -u C  e.  ZZ )
1615adantr 270 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  -u C  e.  ZZ )
17 simpr 108 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  ( C  +  B
) )
18 simpl3r 995 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  C )
19 simpl3l 994 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  C  e.  ZZ )
20 dvdsnegb 10420 . . . . . 6  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  C  <->  A 
||  -u C ) )
219, 19, 20syl2anc 403 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  ( A  ||  C  <->  A  ||  -u C
) )
2218, 21mpbid 145 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  -u C )
23 dvds2add 10437 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  -u C  e.  ZZ )  ->  ( ( A 
||  ( C  +  B )  /\  A  ||  -u C )  ->  A  ||  ( ( C  +  B )  +  -u C ) ) )
2423imp 122 . . . 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 1178 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  ( ( C  +  B )  +  -u C ) )
26 simpl2 943 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  B  e.  ZZ )
2712ancoms 264 . . . . . . 7  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( C  +  B
)  e.  ZZ )
2827zcnd 8603 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( C  +  B
)  e.  CC )
29 zcn 8489 . . . . . . 7  |-  ( C  e.  ZZ  ->  C  e.  CC )
3029adantl 271 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  C  e.  CC )
3128, 30negsubd 7544 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  +  -u C )  =  ( ( C  +  B
)  -  C ) )
32 zcn 8489 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
3332adantr 270 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  B  e.  CC )
3430, 33pncan2d 7540 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  -  C
)  =  B )
3531, 34eqtrd 2115 . . . 4  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  +  -u C )  =  B )
3626, 19, 35syl2anc 403 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  (
( C  +  B
)  +  -u C
)  =  B )
3725, 36breqtrd 3829 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  B )
388, 37impbida 561 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 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3805  (class class class)co 5563   CCcc 7093    + caddc 7098    - cmin 7398   -ucneg 7399   ZZcz 8484    || cdvds 10403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-dvds 10404
This theorem is referenced by:  3dvdsdec  10472  3dvds2dec  10473
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