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Theorem mulqaddmodid 10149
Description: The sum of a positive rational number less than an upper bound and the product of an integer and the upper bound is the positive rational number modulo the upper bound. (Contributed by Jim Kingdon, 23-Oct-2021.)
Assertion
Ref Expression
mulqaddmodid  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
( ( ( N  x.  M )  +  A )  mod  M
)  =  A )

Proof of Theorem mulqaddmodid
StepHypRef Expression
1 simpll 518 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  ->  N  e.  ZZ )
21zcnd 9186 . . . . 5  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  ->  N  e.  CC )
3 qre 9429 . . . . . . 7  |-  ( M  e.  QQ  ->  M  e.  RR )
43ad2antlr 480 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  ->  M  e.  RR )
54recnd 7806 . . . . 5  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  ->  M  e.  CC )
62, 5mulcld 7798 . . . 4  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
( N  x.  M
)  e.  CC )
7 simprr 521 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  ->  A  e.  ( 0 [,) M ) )
8 0red 7779 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
0  e.  RR )
94rexrd 7827 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  ->  M  e.  RR* )
10 elico2 9732 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  M  e.  RR* )  -> 
( A  e.  ( 0 [,) M )  <-> 
( A  e.  RR  /\  0  <_  A  /\  A  <  M ) ) )
118, 9, 10syl2anc 408 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
( A  e.  ( 0 [,) M )  <-> 
( A  e.  RR  /\  0  <_  A  /\  A  <  M ) ) )
127, 11mpbid 146 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
( A  e.  RR  /\  0  <_  A  /\  A  <  M ) )
1312simp1d 993 . . . . 5  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  ->  A  e.  RR )
1413recnd 7806 . . . 4  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  ->  A  e.  CC )
156, 14addcomd 7925 . . 3  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
( ( N  x.  M )  +  A
)  =  ( A  +  ( N  x.  M ) ) )
1615oveq1d 5789 . 2  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
( ( ( N  x.  M )  +  A )  mod  M
)  =  ( ( A  +  ( N  x.  M ) )  mod  M ) )
17 simprl 520 . . 3  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  ->  A  e.  QQ )
18 simplr 519 . . 3  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  ->  M  e.  QQ )
1912simp2d 994 . . . 4  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
0  <_  A )
2012simp3d 995 . . . 4  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  ->  A  <  M )
218, 13, 4, 19, 20lelttrd 7899 . . 3  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
0  <  M )
22 modqcyc 10144 . . 3  |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( A  +  ( N  x.  M
) )  mod  M
)  =  ( A  mod  M ) )
2317, 1, 18, 21, 22syl22anc 1217 . 2  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
( ( A  +  ( N  x.  M
) )  mod  M
)  =  ( A  mod  M ) )
24 modqid 10134 . . 3  |-  ( ( ( A  e.  QQ  /\  M  e.  QQ )  /\  ( 0  <_  A  /\  A  <  M
) )  ->  ( A  mod  M )  =  A )
2517, 18, 19, 20, 24syl22anc 1217 . 2  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
( A  mod  M
)  =  A )
2616, 23, 253eqtrd 2176 1  |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
( ( ( N  x.  M )  +  A )  mod  M
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   RRcr 7631   0cc0 7632    + caddc 7635    x. cmul 7637   RR*cxr 7811    < clt 7812    <_ cle 7813   ZZcz 9066   QQcq 9423   [,)cico 9685    mod cmo 10107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-n0 8990  df-z 9067  df-q 9424  df-rp 9454  df-ico 9689  df-fl 10055  df-mod 10108
This theorem is referenced by:  modqmuladd  10151  addmodid  10157
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