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Theorem qnegmod 9983
Description: The negation of a number modulo a positive number is equal to the difference of the modulus and the number modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
Assertion
Ref Expression
qnegmod  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  ( -u A  mod  N )  =  ( ( N  -  A )  mod 
N ) )

Proof of Theorem qnegmod
StepHypRef Expression
1 qcn 9276 . . . . . 6  |-  ( N  e.  QQ  ->  N  e.  CC )
213ad2ant2 971 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  N  e.  CC )
3 qcn 9276 . . . . . 6  |-  ( A  e.  QQ  ->  A  e.  CC )
433ad2ant1 970 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  A  e.  CC )
52, 4negsubd 7950 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  ( N  +  -u A )  =  ( N  -  A ) )
65eqcomd 2105 . . 3  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  ( N  -  A )  =  ( N  +  -u A ) )
76oveq1d 5721 . 2  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( N  -  A
)  mod  N )  =  ( ( N  +  -u A )  mod 
N ) )
82mulid2d 7656 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
1  x.  N )  =  N )
98oveq1d 5721 . . 3  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( 1  x.  N
)  +  -u A
)  =  ( N  +  -u A ) )
109oveq1d 5721 . 2  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( ( 1  x.  N )  +  -u A )  mod  N
)  =  ( ( N  +  -u A
)  mod  N )
)
11 1cnd 7654 . . . . . 6  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  1  e.  CC )
1211, 2mulcld 7658 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
1  x.  N )  e.  CC )
13 qnegcl 9278 . . . . . . 7  |-  ( A  e.  QQ  ->  -u A  e.  QQ )
14 qcn 9276 . . . . . . 7  |-  ( -u A  e.  QQ  ->  -u A  e.  CC )
1513, 14syl 14 . . . . . 6  |-  ( A  e.  QQ  ->  -u A  e.  CC )
16153ad2ant1 970 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  -u A  e.  CC )
1712, 16addcomd 7784 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( 1  x.  N
)  +  -u A
)  =  ( -u A  +  ( 1  x.  N ) ) )
1817oveq1d 5721 . . 3  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( ( 1  x.  N )  +  -u A )  mod  N
)  =  ( (
-u A  +  ( 1  x.  N ) )  mod  N ) )
19133ad2ant1 970 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  -u A  e.  QQ )
20 1zzd 8933 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  1  e.  ZZ )
21 simp2 950 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  N  e.  QQ )
22 simp3 951 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  0  <  N )
23 modqcyc 9973 . . . 4  |-  ( ( ( -u A  e.  QQ  /\  1  e.  ZZ )  /\  ( N  e.  QQ  /\  0  <  N ) )  -> 
( ( -u A  +  ( 1  x.  N ) )  mod 
N )  =  (
-u A  mod  N
) )
2419, 20, 21, 22, 23syl22anc 1185 . . 3  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( -u A  +  ( 1  x.  N ) )  mod  N )  =  ( -u A  mod  N ) )
2518, 24eqtrd 2132 . 2  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( ( 1  x.  N )  +  -u A )  mod  N
)  =  ( -u A  mod  N ) )
267, 10, 253eqtr2rd 2139 1  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  ( -u A  mod  N )  =  ( ( N  -  A )  mod 
N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 930    = wceq 1299    e. wcel 1448   class class class wbr 3875  (class class class)co 5706   CCcc 7498   0cc0 7500   1c1 7501    + caddc 7503    x. cmul 7505    < clt 7672    - cmin 7804   -ucneg 7805   ZZcz 8906   QQcq 9261    mod cmo 9936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-n0 8830  df-z 8907  df-q 9262  df-rp 9292  df-fl 9884  df-mod 9937
This theorem is referenced by:  m1modnnsub1  9984
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