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Theorem addmodid 9744
Description: The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by Alexander van der Vekens, 30-Oct-2018.) (Proof shortened by AV, 5-Jul-2020.)
Assertion
Ref Expression
addmodid  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  (
( M  +  A
)  mod  M )  =  A )

Proof of Theorem addmodid
StepHypRef Expression
1 simp2 944 . . . . . . 7  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  M  e.  NN )
21nncnd 8408 . . . . . 6  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  M  e.  CC )
32mulid2d 7485 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  (
1  x.  M )  =  M )
43eqcomd 2093 . . . 4  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  M  =  ( 1  x.  M ) )
54oveq1d 5649 . . 3  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  ( M  +  A )  =  ( ( 1  x.  M )  +  A ) )
65oveq1d 5649 . 2  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  (
( M  +  A
)  mod  M )  =  ( ( ( 1  x.  M )  +  A )  mod 
M ) )
7 1zzd 8747 . . 3  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  1  e.  ZZ )
8 nnq 9087 . . . 4  |-  ( M  e.  NN  ->  M  e.  QQ )
983ad2ant2 965 . . 3  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  M  e.  QQ )
10 simp1 943 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  A  e.  NN0 )
1110nn0zd 8836 . . . 4  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  A  e.  ZZ )
12 zq 9080 . . . 4  |-  ( A  e.  ZZ  ->  A  e.  QQ )
1311, 12syl 14 . . 3  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  A  e.  QQ )
14 nn0re 8652 . . . . 5  |-  ( A  e.  NN0  ->  A  e.  RR )
15143ad2ant1 964 . . . 4  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  A  e.  RR )
1610nn0ge0d 8699 . . . 4  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  0  <_  A )
17 simp3 945 . . . 4  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  A  <  M )
18 0re 7467 . . . . 5  |-  0  e.  RR
19 nnre 8401 . . . . . . 7  |-  ( M  e.  NN  ->  M  e.  RR )
2019rexrd 7516 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  RR* )
21203ad2ant2 965 . . . . 5  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  M  e.  RR* )
22 elico2 9324 . . . . 5  |-  ( ( 0  e.  RR  /\  M  e.  RR* )  -> 
( A  e.  ( 0 [,) M )  <-> 
( A  e.  RR  /\  0  <_  A  /\  A  <  M ) ) )
2318, 21, 22sylancr 405 . . . 4  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  ( A  e.  ( 0 [,) M )  <->  ( A  e.  RR  /\  0  <_  A  /\  A  <  M
) ) )
2415, 16, 17, 23mpbir3and 1126 . . 3  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  A  e.  ( 0 [,) M
) )
25 mulqaddmodid 9736 . . 3  |-  ( ( ( 1  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M
) ) )  -> 
( ( ( 1  x.  M )  +  A )  mod  M
)  =  A )
267, 9, 13, 24, 25syl22anc 1175 . 2  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  (
( ( 1  x.  M )  +  A
)  mod  M )  =  A )
276, 26eqtrd 2120 1  |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  (
( M  +  A
)  mod  M )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   RRcr 7328   0cc0 7329   1c1 7330    + caddc 7332    x. cmul 7334   RR*cxr 7500    < clt 7501    <_ cle 7502   NNcn 8394   NN0cn0 8643   ZZcz 8720   QQcq 9073   [,)cico 9277    mod cmo 9694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-n0 8644  df-z 8721  df-q 9074  df-rp 9104  df-ico 9281  df-fl 9642  df-mod 9695
This theorem is referenced by:  addmodidr  9745
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