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Theorem dvdsadd2b 13596
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 991 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  e.  ZZ )
2 simpl3l 1043 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  C  e.  ZZ )
3 simpl2 992 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  B  e.  ZZ )
4 simpl3r 1044 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  C )
5 simpr 461 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  B )
6 dvds2add 13585 . . . 4  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  ||  C  /\  A  ||  B )  ->  A  ||  ( C  +  B )
) )
76imp 429 . . 3  |-  ( ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  /\  ( A  ||  C  /\  A  ||  B ) )  ->  A  ||  ( C  +  B )
)
81, 2, 3, 4, 5, 7syl32anc 1226 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  B )  ->  A  ||  ( C  +  B
) )
9 simpl1 991 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  e.  ZZ )
10 simp3l 1016 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  C  e.  ZZ )
11 simp2 989 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  B  e.  ZZ )
12 zaddcl 10706 . . . . . 6  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  +  B
)  e.  ZZ )
1310, 11, 12syl2anc 661 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  -> 
( C  +  B
)  e.  ZZ )
1413adantr 465 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  ( C  +  B )  e.  ZZ )
1510znegcld 10770 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  ->  -u C  e.  ZZ )
1615adantr 465 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  -u C  e.  ZZ )
17 simpr 461 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  ( C  +  B
) )
18 simpl3r 1044 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  C )
19 simpl3l 1043 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  C  e.  ZZ )
20 dvdsnegb 13571 . . . . . 6  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  C  <->  A 
||  -u C ) )
219, 19, 20syl2anc 661 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  ( A  ||  C  <->  A  ||  -u C
) )
2218, 21mpbid 210 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  -u C )
23 dvds2add 13585 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  -u C  e.  ZZ )  ->  ( ( A 
||  ( C  +  B )  /\  A  ||  -u C )  ->  A  ||  ( ( C  +  B )  +  -u C ) ) )
2423imp 429 . . . 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 1226 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  ( ( C  +  B )  +  -u C ) )
26 simpl2 992 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  B  e.  ZZ )
2712ancoms 453 . . . . . . 7  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( C  +  B
)  e.  ZZ )
2827zcnd 10769 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( C  +  B
)  e.  CC )
29 zcn 10672 . . . . . . 7  |-  ( C  e.  ZZ  ->  C  e.  CC )
3029adantl 466 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  C  e.  CC )
3128, 30negsubd 9746 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  +  -u C )  =  ( ( C  +  B
)  -  C ) )
32 zcn 10672 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
3332adantr 465 . . . . . 6  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  B  e.  CC )
3430, 33pncan2d 9742 . . . . 5  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  -  C
)  =  B )
3531, 34eqtrd 2475 . . . 4  |-  ( ( B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( C  +  B )  +  -u C )  =  B )
3626, 19, 35syl2anc 661 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  (
( C  +  B
)  +  -u C
)  =  B )
3725, 36breqtrd 4337 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  /\  A  ||  ( C  +  B
) )  ->  A  ||  B )
388, 37impbida 828 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( C  e.  ZZ  /\  A  ||  C ) )  -> 
( A  ||  B  <->  A 
||  ( C  +  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4313  (class class class)co 6112   CCcc 9301    + caddc 9306    - cmin 9616   -ucneg 9617   ZZcz 10667    || cdivides 13556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-n0 10601  df-z 10668  df-dvds 13557
This theorem is referenced by:  2sqlem3  22727  2sqblem  22738  eupath2lem3  23622  jm2.19lem2  29365
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