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Theorem modabsdifz 29259
Description: Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 26-Sep-2014.)
Assertion
Ref Expression
modabsdifz  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  (
( N  -  ( N  mod  ( abs `  M
) ) )  /  M )  e.  ZZ )

Proof of Theorem modabsdifz
StepHypRef Expression
1 simp1 983 . . 3  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  N  e.  RR )
2 simp2 984 . . . . 5  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  M  e.  RR )
32recnd 9408 . . . 4  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  M  e.  CC )
4 simp3 985 . . . 4  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  M  =/=  0 )
53, 4absrpcld 12930 . . 3  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( abs `  M )  e.  RR+ )
6 moddifz 11716 . . 3  |-  ( ( N  e.  RR  /\  ( abs `  M )  e.  RR+ )  ->  (
( N  -  ( N  mod  ( abs `  M
) ) )  / 
( abs `  M
) )  e.  ZZ )
71, 5, 6syl2anc 656 . 2  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  (
( N  -  ( N  mod  ( abs `  M
) ) )  / 
( abs `  M
) )  e.  ZZ )
8 absidm 12807 . . . . . . 7  |-  ( M  e.  CC  ->  ( abs `  ( abs `  M
) )  =  ( abs `  M ) )
93, 8syl 16 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( abs `  ( abs `  M
) )  =  ( abs `  M ) )
109oveq2d 6106 . . . . 5  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  (
( abs `  ( N  -  ( N  mod  ( abs `  M
) ) ) )  /  ( abs `  ( abs `  M ) ) )  =  ( ( abs `  ( N  -  ( N  mod  ( abs `  M ) ) ) )  / 
( abs `  M
) ) )
111, 5modcld 11710 . . . . . . . 8  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( N  mod  ( abs `  M
) )  e.  RR )
121, 11resubcld 9772 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( N  -  ( N  mod  ( abs `  M
) ) )  e.  RR )
1312recnd 9408 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( N  -  ( N  mod  ( abs `  M
) ) )  e.  CC )
143abscld 12918 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( abs `  M )  e.  RR )
1514recnd 9408 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( abs `  M )  e.  CC )
165rpne0d 11028 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( abs `  M )  =/=  0 )
1713, 15, 16absdivd 12937 . . . . 5  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( abs `  ( ( N  -  ( N  mod  ( abs `  M ) ) )  /  ( abs `  M ) ) )  =  ( ( abs `  ( N  -  ( N  mod  ( abs `  M ) ) ) )  / 
( abs `  ( abs `  M ) ) ) )
1813, 3, 4absdivd 12937 . . . . 5  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( abs `  ( ( N  -  ( N  mod  ( abs `  M ) ) )  /  M
) )  =  ( ( abs `  ( N  -  ( N  mod  ( abs `  M
) ) ) )  /  ( abs `  M
) ) )
1910, 17, 183eqtr4d 2483 . . . 4  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  ( abs `  ( ( N  -  ( N  mod  ( abs `  M ) ) )  /  ( abs `  M ) ) )  =  ( abs `  ( ( N  -  ( N  mod  ( abs `  M ) ) )  /  M ) ) )
2019eleq1d 2507 . . 3  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  (
( abs `  (
( N  -  ( N  mod  ( abs `  M
) ) )  / 
( abs `  M
) ) )  e.  ZZ  <->  ( abs `  (
( N  -  ( N  mod  ( abs `  M
) ) )  /  M ) )  e.  ZZ ) )
2112, 14, 16redivcld 10155 . . . 4  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  (
( N  -  ( N  mod  ( abs `  M
) ) )  / 
( abs `  M
) )  e.  RR )
22 absz 12796 . . . 4  |-  ( ( ( N  -  ( N  mod  ( abs `  M
) ) )  / 
( abs `  M
) )  e.  RR  ->  ( ( ( N  -  ( N  mod  ( abs `  M ) ) )  /  ( abs `  M ) )  e.  ZZ  <->  ( abs `  ( ( N  -  ( N  mod  ( abs `  M ) ) )  /  ( abs `  M
) ) )  e.  ZZ ) )
2321, 22syl 16 . . 3  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  (
( ( N  -  ( N  mod  ( abs `  M ) ) )  /  ( abs `  M
) )  e.  ZZ  <->  ( abs `  ( ( N  -  ( N  mod  ( abs `  M
) ) )  / 
( abs `  M
) ) )  e.  ZZ ) )
2412, 2, 4redivcld 10155 . . . 4  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  (
( N  -  ( N  mod  ( abs `  M
) ) )  /  M )  e.  RR )
25 absz 12796 . . . 4  |-  ( ( ( N  -  ( N  mod  ( abs `  M
) ) )  /  M )  e.  RR  ->  ( ( ( N  -  ( N  mod  ( abs `  M ) ) )  /  M
)  e.  ZZ  <->  ( abs `  ( ( N  -  ( N  mod  ( abs `  M ) ) )  /  M ) )  e.  ZZ ) )
2624, 25syl 16 . . 3  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  (
( ( N  -  ( N  mod  ( abs `  M ) ) )  /  M )  e.  ZZ  <->  ( abs `  (
( N  -  ( N  mod  ( abs `  M
) ) )  /  M ) )  e.  ZZ ) )
2720, 23, 263bitr4d 285 . 2  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  (
( ( N  -  ( N  mod  ( abs `  M ) ) )  /  ( abs `  M
) )  e.  ZZ  <->  ( ( N  -  ( N  mod  ( abs `  M
) ) )  /  M )  e.  ZZ ) )
287, 27mpbid 210 1  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  M  =/=  0 )  ->  (
( N  -  ( N  mod  ( abs `  M
) ) )  /  M )  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278    - cmin 9591    / cdiv 9989   ZZcz 10642   RR+crp 10987    mod cmo 11704   abscabs 12719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721
This theorem is referenced by:  jm2.19  29267
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