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Theorem modqmuladdim 9739
Description: Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
Assertion
Ref Expression
modqmuladdim  |-  ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  mod  M
)  =  B  ->  E. k  e.  ZZ  A  =  ( (
k  x.  M )  +  B ) ) )
Distinct variable groups:    A, k    B, k    k, M

Proof of Theorem modqmuladdim
StepHypRef Expression
1 simpr 108 . . 3  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  mod  M
)  =  B )
2 simpl1 946 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  A  e.  ZZ )
3 zq 9080 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  QQ )
42, 3syl 14 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  A  e.  QQ )
5 simpl2 947 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  M  e.  QQ )
6 simpl3 948 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
0  <  M )
74, 5, 6modqcld 9700 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  mod  M
)  e.  QQ )
81, 7eqeltrrd 2165 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  B  e.  QQ )
9 qre 9079 . . . . . 6  |-  ( B  e.  QQ  ->  B  e.  RR )
108, 9syl 14 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  B  e.  RR )
11 modqge0 9704 . . . . . . 7  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  0  <_  ( A  mod  M
) )
124, 5, 6, 11syl3anc 1174 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
0  <_  ( A  mod  M ) )
1312, 1breqtrd 3861 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
0  <_  B )
14 modqlt 9705 . . . . . . 7  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  ( A  mod  M )  < 
M )
154, 5, 6, 14syl3anc 1174 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  mod  M
)  <  M )
161, 15eqbrtrrd 3859 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  B  <  M )
17 0re 7467 . . . . . 6  |-  0  e.  RR
18 qre 9079 . . . . . . 7  |-  ( M  e.  QQ  ->  M  e.  RR )
19 rexr 7512 . . . . . . 7  |-  ( M  e.  RR  ->  M  e.  RR* )
205, 18, 193syl 17 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  M  e.  RR* )
21 elico2 9324 . . . . . 6  |-  ( ( 0  e.  RR  /\  M  e.  RR* )  -> 
( B  e.  ( 0 [,) M )  <-> 
( B  e.  RR  /\  0  <_  B  /\  B  <  M ) ) )
2217, 20, 21sylancr 405 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( B  e.  ( 0 [,) M )  <-> 
( B  e.  RR  /\  0  <_  B  /\  B  <  M ) ) )
2310, 13, 16, 22mpbir3and 1126 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  B  e.  ( 0 [,) M ) )
242, 8, 23, 5, 6modqmuladd 9738 . . 3  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( ( A  mod  M )  =  B  <->  E. k  e.  ZZ  A  =  ( ( k  x.  M
)  +  B ) ) )
251, 24mpbid 145 . 2  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  E. k  e.  ZZ  A  =  ( (
k  x.  M )  +  B ) )
2625ex 113 1  |-  ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( A  mod  M
)  =  B  ->  E. k  e.  ZZ  A  =  ( (
k  x.  M )  +  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360   class class class wbr 3837  (class class class)co 5634   RRcr 7328   0cc0 7329    + caddc 7332    x. cmul 7334   RR*cxr 7500    < clt 7501    <_ cle 7502   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:  modqmuladdnn0  9740
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