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Theorem dvdsadd2b 12892
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 960 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  e.  ZZ )
2 simpl3l 1012 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  C  e.  ZZ )
3 simpl2 961 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  B  e.  ZZ )
4 simpl3r 1013 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  C )
5 simpr 448 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  B )
6 dvds2add 12881 . . . 4  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  ||  C  /\  A  ||  B )  ->  A  ||  ( C  +  B )
) )
76imp 419 . . 3  |-  ( ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  ||  C  /\  A  ||  B ) )  ->  A  ||  ( C  +  B )
)
81, 2, 3, 4, 5, 7syl32anc 1192 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  ( C  +  B
) )
9 simpl1 960 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  e.  ZZ )
10 simp3l 985 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  C  e.  ZZ )
11 simp2 958 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  B  e.  ZZ )
12 zaddcl 10317 . . . . . 6  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  +  B
)  e.  ZZ )
1310, 11, 12syl2anc 643 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  -> 
( C  +  B
)  e.  ZZ )
1413adantr 452 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  ( C  +  B )  e.  ZZ )
1510znegcld 10377 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  -u C  e.  ZZ )
1615adantr 452 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  -u C  e.  ZZ )
17 simpr 448 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  ( C  +  B
) )
18 simpl3r 1013 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  C )
19 simpl3l 1012 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  C  e.  ZZ )
20 dvdsnegb 12867 . . . . . 6  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  C  <->  A 
||  -u C ) )
219, 19, 20syl2anc 643 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  ( A  ||  C  <->  A  ||  -u C
) )
2218, 21mpbid 202 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  -u C )
23 dvds2add 12881 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  -u C  e.  ZZ )  ->  ( ( A 
||  ( C  +  B )  /\  A  ||  -u C )  ->  A  ||  ( ( C  +  B )  +  -u C ) ) )
2423imp 419 . . . 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 1192 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  ( ( C  +  B )  +  -u C ) )
26 simpl2 961 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  B  e.  ZZ )
2712ancoms 440 . . . . . . 7  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( C  +  B
)  e.  ZZ )
2827zcnd 10376 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( C  +  B
)  e.  CC )
29 zcn 10287 . . . . . . 7  |-  ( C  e.  ZZ  ->  C  e.  CC )
3029adantl 453 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  C  e.  CC )
3128, 30negsubd 9417 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  +  -u C )  =  ( ( C  +  B
)  -  C ) )
32 zcn 10287 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
3332adantr 452 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  B  e.  CC )
3430, 33pncan2d 9413 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  -  C
)  =  B )
3531, 34eqtrd 2468 . . . 4  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  +  -u C )  =  B )
3626, 19, 35syl2anc 643 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  (
( C  +  B
)  +  -u C
)  =  B )
3725, 36breqtrd 4236 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  B )
388, 37impbida 806 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    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   CCcc 8988    + caddc 8993    - cmin 9291   -ucneg 9292   ZZcz 10282    || cdivides 12852
This theorem is referenced by:  2sqlem3  21150  2sqblem  21161  eupath2lem3  21701  jm2.19lem2  27061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-dvds 12853
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