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Theorem negmod0 10979
Description:  A is divisible by  B iff its negative is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)
Assertion
Ref Expression
negmod0  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( -u A  mod  B )  =  0 ) )

Proof of Theorem negmod0
StepHypRef Expression
1 rerpdivcl 10381 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
2 recn 8827 . . . 4  |-  ( ( A  /  B )  e.  RR  ->  ( A  /  B )  e.  CC )
3 znegclb 10056 . . . 4  |-  ( ( A  /  B )  e.  CC  ->  (
( A  /  B
)  e.  ZZ  <->  -u ( A  /  B )  e.  ZZ ) )
41, 2, 33syl 18 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  e.  ZZ  <->  -u ( A  /  B
)  e.  ZZ ) )
5 recn 8827 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
65adantr 451 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  A  e.  CC )
7 rpcn 10362 . . . . . 6  |-  ( B  e.  RR+  ->  B  e.  CC )
87adantl 452 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  CC )
9 rpne0 10369 . . . . . 6  |-  ( B  e.  RR+  ->  B  =/=  0 )
109adantl 452 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  =/=  0 )
116, 8, 10divnegd 9549 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  -u ( A  /  B
)  =  ( -u A  /  B ) )
1211eleq1d 2349 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( -u ( A  /  B )  e.  ZZ  <->  (
-u A  /  B
)  e.  ZZ ) )
134, 12bitrd 244 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  e.  ZZ  <->  (
-u A  /  B
)  e.  ZZ ) )
14 mod0 10978 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( A  /  B )  e.  ZZ ) )
15 renegcl 9110 . . 3  |-  ( A  e.  RR  ->  -u A  e.  RR )
16 mod0 10978 . . 3  |-  ( (
-u A  e.  RR  /\  B  e.  RR+ )  ->  ( ( -u A  mod  B )  =  0  <-> 
( -u A  /  B
)  e.  ZZ ) )
1715, 16sylan 457 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( -u A  mod  B )  =  0  <-> 
( -u A  /  B
)  e.  ZZ ) )
1813, 14, 173bitr4d 276 1  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  mod  B )  =  0  <->  ( -u A  mod  B )  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   -ucneg 9038    / cdiv 9423   ZZcz 10024   RR+crp 10354    mod cmo 10973
This theorem is referenced by:  absmod0  11788  negmod0OLD  26450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974
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