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Theorem dvdsadd2b 11270
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 949 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  e.  ZZ )
2 simpl3l 1001 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  C  e.  ZZ )
3 simpl2 950 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  B  e.  ZZ )
4 simpl3r 1002 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  C )
5 simpr 109 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  B )
6 dvds2add 11257 . . . 4  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  ||  C  /\  A  ||  B )  ->  A  ||  ( C  +  B )
) )
76imp 123 . . 3  |-  ( ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  ||  C  /\  A  ||  B ) )  ->  A  ||  ( C  +  B )
)
81, 2, 3, 4, 5, 7syl32anc 1189 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  ( C  +  B
) )
9 simpl1 949 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  e.  ZZ )
10 simp3l 974 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  C  e.  ZZ )
11 simp2 947 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  B  e.  ZZ )
12 zaddcl 8888 . . . . . 6  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  +  B
)  e.  ZZ )
1310, 11, 12syl2anc 404 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  -> 
( C  +  B
)  e.  ZZ )
1413adantr 271 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  ( C  +  B )  e.  ZZ )
1510znegcld 8969 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  -u C  e.  ZZ )
1615adantr 271 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  -u C  e.  ZZ )
17 simpr 109 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  ( C  +  B
) )
18 simpl3r 1002 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  C )
19 simpl3l 1001 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  C  e.  ZZ )
20 dvdsnegb 11240 . . . . . 6  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  C  <->  A 
||  -u C ) )
219, 19, 20syl2anc 404 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  ( A  ||  C  <->  A  ||  -u C
) )
2218, 21mpbid 146 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  -u C )
23 dvds2add 11257 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  -u C  e.  ZZ )  ->  ( ( A 
||  ( C  +  B )  /\  A  ||  -u C )  ->  A  ||  ( ( C  +  B )  +  -u C ) ) )
2423imp 123 . . . 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 1189 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  ( ( C  +  B )  +  -u C ) )
26 simpl2 950 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  B  e.  ZZ )
2712ancoms 265 . . . . . . 7  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( C  +  B
)  e.  ZZ )
2827zcnd 8968 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( C  +  B
)  e.  CC )
29 zcn 8853 . . . . . . 7  |-  ( C  e.  ZZ  ->  C  e.  CC )
3029adantl 272 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  C  e.  CC )
3128, 30negsubd 7896 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  +  -u C )  =  ( ( C  +  B
)  -  C ) )
32 zcn 8853 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
3332adantr 271 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  B  e.  CC )
3430, 33pncan2d 7892 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  -  C
)  =  B )
3531, 34eqtrd 2127 . . . 4  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  +  -u C )  =  B )
3626, 19, 35syl2anc 404 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  (
( C  +  B
)  +  -u C
)  =  B )
3725, 36breqtrd 3891 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  B )
388, 37impbida 564 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 103    <-> wb 104    /\ w3a 927    = wceq 1296    e. wcel 1445   class class class wbr 3867  (class class class)co 5690   CCcc 7445    + caddc 7450    - cmin 7750   -ucneg 7751   ZZcz 8848    || cdvds 11223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-0id 7550  ax-rnegex 7551  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-ltadd 7558
This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-inn 8521  df-n0 8772  df-z 8849  df-dvds 11224
This theorem is referenced by:  3dvdsdec  11292  3dvds2dec  11293
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