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Theorem qnegmod 10399
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 9663 . . . . . 6  |-  ( N  e.  QQ  ->  N  e.  CC )
213ad2ant2 1021 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  N  e.  CC )
3 qcn 9663 . . . . . 6  |-  ( A  e.  QQ  ->  A  e.  CC )
433ad2ant1 1020 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  A  e.  CC )
52, 4negsubd 8303 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  ( N  +  -u A )  =  ( N  -  A ) )
65eqcomd 2195 . . 3  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  ( N  -  A )  =  ( N  +  -u A ) )
76oveq1d 5910 . 2  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( N  -  A
)  mod  N )  =  ( ( N  +  -u A )  mod 
N ) )
82mulid2d 8005 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
1  x.  N )  =  N )
98oveq1d 5910 . . 3  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( 1  x.  N
)  +  -u A
)  =  ( N  +  -u A ) )
109oveq1d 5910 . 2  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( ( 1  x.  N )  +  -u A )  mod  N
)  =  ( ( N  +  -u A
)  mod  N )
)
11 1cnd 8002 . . . . . 6  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  1  e.  CC )
1211, 2mulcld 8007 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
1  x.  N )  e.  CC )
13 qnegcl 9665 . . . . . . 7  |-  ( A  e.  QQ  ->  -u A  e.  QQ )
14 qcn 9663 . . . . . . 7  |-  ( -u A  e.  QQ  ->  -u A  e.  CC )
1513, 14syl 14 . . . . . 6  |-  ( A  e.  QQ  ->  -u A  e.  CC )
16153ad2ant1 1020 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  -u A  e.  CC )
1712, 16addcomd 8137 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( 1  x.  N
)  +  -u A
)  =  ( -u A  +  ( 1  x.  N ) ) )
1817oveq1d 5910 . . 3  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( ( 1  x.  N )  +  -u A )  mod  N
)  =  ( (
-u A  +  ( 1  x.  N ) )  mod  N ) )
19133ad2ant1 1020 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  -u A  e.  QQ )
20 1zzd 9309 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  1  e.  ZZ )
21 simp2 1000 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  N  e.  QQ )
22 simp3 1001 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  0  <  N )
23 modqcyc 10389 . . . 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 1250 . . 3  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( -u A  +  ( 1  x.  N ) )  mod  N )  =  ( -u A  mod  N ) )
2518, 24eqtrd 2222 . 2  |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  (
( ( 1  x.  N )  +  -u A )  mod  N
)  =  ( -u A  mod  N ) )
267, 10, 253eqtr2rd 2229 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 980    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5895   CCcc 7838   0cc0 7840   1c1 7841    + caddc 7843    x. cmul 7845    < clt 8021    - cmin 8157   -ucneg 8158   ZZcz 9282   QQcq 9648    mod cmo 10352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-n0 9206  df-z 9283  df-q 9649  df-rp 9683  df-fl 10300  df-mod 10353
This theorem is referenced by:  m1modnnsub1  10400
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