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Theorem modqmuladdim 10364
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 110 . . 3  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  mod  M
)  =  B )
2 simpl1 1000 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  A  e.  ZZ )
3 zq 9624 . . . . . . 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 1001 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  M  e.  QQ )
6 simpl3 1002 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
0  <  M )
74, 5, 6modqcld 10325 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  mod  M
)  e.  QQ )
81, 7eqeltrrd 2255 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  B  e.  QQ )
9 qre 9623 . . . . . 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 10329 . . . . . . 7  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  0  <_  ( A  mod  M
) )
124, 5, 6, 11syl3anc 1238 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
0  <_  ( A  mod  M ) )
1312, 1breqtrd 4029 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
0  <_  B )
14 modqlt 10330 . . . . . . 7  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  ( A  mod  M )  < 
M )
154, 5, 6, 14syl3anc 1238 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  -> 
( A  mod  M
)  <  M )
161, 15eqbrtrrd 4027 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  B  <  M )
17 0re 7956 . . . . . 6  |-  0  e.  RR
18 qre 9623 . . . . . . 7  |-  ( M  e.  QQ  ->  M  e.  RR )
19 rexr 8001 . . . . . . 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 9935 . . . . . 6  |-  ( ( 0  e.  RR  /\  M  e.  RR* )  -> 
( B  e.  ( 0 [,) M )  <-> 
( B  e.  RR  /\  0  <_  B  /\  B  <  M ) ) )
2217, 20, 21sylancr 414 . . . . 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 1180 . . . 4  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  B  e.  ( 0 [,) M ) )
242, 8, 23, 5, 6modqmuladd 10363 . . 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 147 . 2  |-  ( ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M )  /\  ( A  mod  M )  =  B )  ->  E. k  e.  ZZ  A  =  ( (
k  x.  M )  +  B ) )
2625ex 115 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456   class class class wbr 4003  (class class class)co 5874   RRcr 7809   0cc0 7810    + caddc 7813    x. cmul 7815   RR*cxr 7989    < clt 7990    <_ cle 7991   ZZcz 9251   QQcq 9617   [,)cico 9888    mod cmo 10319
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-po 4296  df-iso 4297  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-n0 9175  df-z 9252  df-q 9618  df-rp 9652  df-ico 9892  df-fl 10267  df-mod 10320
This theorem is referenced by:  modqmuladdnn0  10365
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