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Theorem modaddmodlo 10192
Description: The sum of an integer modulo a positive integer and another integer equals the sum of the two integers modulo the positive integer if the other integer is in the lower part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
Assertion
Ref Expression
modaddmodlo  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  ( B  +  ( A  mod  M ) )  =  ( ( B  +  A )  mod  M
) ) )

Proof of Theorem modaddmodlo
StepHypRef Expression
1 elfzoelz 9955 . . . . . . 7  |-  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  B  e.  ZZ )
21adantl 275 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  B  e.  ZZ )
3 zq 9445 . . . . . 6  |-  ( B  e.  ZZ  ->  B  e.  QQ )
42, 3syl 14 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  B  e.  QQ )
5 zmodcl 10148 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( A  mod  M
)  e.  NN0 )
65adantr 274 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( A  mod  M
)  e.  NN0 )
76nn0zd 9195 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( A  mod  M
)  e.  ZZ )
8 zq 9445 . . . . . 6  |-  ( ( A  mod  M )  e.  ZZ  ->  ( A  mod  M )  e.  QQ )
97, 8syl 14 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( A  mod  M
)  e.  QQ )
10 qaddcl 9454 . . . . 5  |-  ( ( B  e.  QQ  /\  ( A  mod  M )  e.  QQ )  -> 
( B  +  ( A  mod  M ) )  e.  QQ )
114, 9, 10syl2anc 409 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( B  +  ( A  mod  M ) )  e.  QQ )
12 simplr 520 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  M  e.  NN )
13 nnq 9452 . . . . 5  |-  ( M  e.  NN  ->  M  e.  QQ )
1412, 13syl 14 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  M  e.  QQ )
152zred 9197 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  B  e.  RR )
166nn0red 9055 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( A  mod  M
)  e.  RR )
17 elfzole1 9963 . . . . . 6  |-  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  0  <_  B )
1817adantl 275 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
0  <_  B )
196nn0ge0d 9057 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
0  <_  ( A  mod  M ) )
2015, 16, 18, 19addge0d 8308 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
0  <_  ( B  +  ( A  mod  M ) ) )
21 elfzolt2 9964 . . . . . 6  |-  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  B  <  ( M  -  ( A  mod  M ) ) )
2221adantl 275 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  B  <  ( M  -  ( A  mod  M ) ) )
2312nnred 8757 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  M  e.  RR )
2415, 16, 23ltaddsubd 8331 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( ( B  +  ( A  mod  M ) )  <  M  <->  B  <  ( M  -  ( A  mod  M ) ) ) )
2522, 24mpbird 166 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( B  +  ( A  mod  M ) )  <  M )
26 modqid 10153 . . . 4  |-  ( ( ( ( B  +  ( A  mod  M ) )  e.  QQ  /\  M  e.  QQ )  /\  ( 0  <_  ( B  +  ( A  mod  M ) )  /\  ( B  +  ( A  mod  M ) )  <  M ) )  ->  ( ( B  +  ( A  mod  M ) )  mod  M
)  =  ( B  +  ( A  mod  M ) ) )
2711, 14, 20, 25, 26syl22anc 1218 . . 3  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( ( B  +  ( A  mod  M ) )  mod  M )  =  ( B  +  ( A  mod  M ) ) )
28 zq 9445 . . . . 5  |-  ( A  e.  ZZ  ->  A  e.  QQ )
2928ad2antrr 480 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  ->  A  e.  QQ )
3012nngt0d 8788 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
0  <  M )
31 modqadd2mod 10178 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  < 
M ) )  -> 
( ( B  +  ( A  mod  M ) )  mod  M )  =  ( ( B  +  A )  mod 
M ) )
3229, 4, 14, 30, 31syl22anc 1218 . . 3  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( ( B  +  ( A  mod  M ) )  mod  M )  =  ( ( B  +  A )  mod 
M ) )
3327, 32eqtr3d 2175 . 2  |-  ( ( ( A  e.  ZZ  /\  M  e.  NN )  /\  B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) ) )  -> 
( B  +  ( A  mod  M ) )  =  ( ( B  +  A )  mod  M ) )
3433ex 114 1  |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  ( B  +  ( A  mod  M ) )  =  ( ( B  +  A )  mod  M
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   class class class wbr 3937  (class class class)co 5782   0cc0 7644    + caddc 7647    < clt 7824    <_ cle 7825    - cmin 7957   NNcn 8744   NN0cn0 9001   ZZcz 9078   QQcq 9438  ..^cfzo 9950    mod cmo 10126
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-fl 10074  df-mod 10127
This theorem is referenced by: (None)
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