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Theorem dvdsabsmod0 27059
Description: Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014.)
Assertion
Ref Expression
dvdsabsmod0  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  ( M  ||  N  <->  ( N  mod  ( abs `  M
) )  =  0 ) )

Proof of Theorem dvdsabsmod0
StepHypRef Expression
1 absdvdsb 12870 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( abs `  M ) 
||  N ) )
213adant3 978 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  ( M  ||  N  <->  ( abs `  M )  ||  N
) )
3 zcn 10289 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
433ad2ant2 980 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  N  e.  CC )
54subid1d 9402 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  ( N  -  0 )  =  N )
65breq2d 4226 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  (
( abs `  M
)  ||  ( N  -  0 )  <->  ( abs `  M )  ||  N
) )
72, 6bitr4d 249 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  ( M  ||  N  <->  ( abs `  M )  ||  ( N  -  0 ) ) )
8 nnabscl 12131 . . . 4  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
983adant2 977 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  ( abs `  M )  e.  NN )
10 simp2 959 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  N  e.  ZZ )
11 0z 10295 . . . 4  |-  0  e.  ZZ
1211a1i 11 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  0  e.  ZZ )
13 moddvds 12861 . . 3  |-  ( ( ( abs `  M
)  e.  NN  /\  N  e.  ZZ  /\  0  e.  ZZ )  ->  (
( N  mod  ( abs `  M ) )  =  ( 0  mod  ( abs `  M
) )  <->  ( abs `  M )  ||  ( N  -  0 ) ) )
149, 10, 12, 13syl3anc 1185 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  (
( N  mod  ( abs `  M ) )  =  ( 0  mod  ( abs `  M
) )  <->  ( abs `  M )  ||  ( N  -  0 ) ) )
15 nnrp 10623 . . . 4  |-  ( ( abs `  M )  e.  NN  ->  ( abs `  M )  e.  RR+ )
16 0mod 11274 . . . 4  |-  ( ( abs `  M )  e.  RR+  ->  ( 0  mod  ( abs `  M
) )  =  0 )
179, 15, 163syl 19 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  (
0  mod  ( abs `  M ) )  =  0 )
1817eqeq2d 2449 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  (
( N  mod  ( abs `  M ) )  =  ( 0  mod  ( abs `  M
) )  <->  ( N  mod  ( abs `  M
) )  =  0 ) )
197, 14, 183bitr2d 274 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  =/=  0 )  ->  ( M  ||  N  <->  ( N  mod  ( abs `  M
) )  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992    - cmin 9293   NNcn 10002   ZZcz 10284   RR+crp 10614    mod cmo 11252   abscabs 12041    || cdivides 12854
This theorem is referenced by:  jm2.19  27066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855
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