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Theorem modqmuladd 10396
Description: Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
Hypotheses
Ref Expression
modqmuladd.a  |-  ( ph  ->  A  e.  ZZ )
modqmuladd.bq  |-  ( ph  ->  B  e.  QQ )
modqmuladd.b  |-  ( ph  ->  B  e.  ( 0 [,) M ) )
modqmuladd.m  |-  ( ph  ->  M  e.  QQ )
modqmuladd.mgt0  |-  ( ph  ->  0  <  M )
Assertion
Ref Expression
modqmuladd  |-  ( ph  ->  ( ( A  mod  M )  =  B  <->  E. k  e.  ZZ  A  =  ( ( k  x.  M
)  +  B ) ) )
Distinct variable groups:    A, k    B, k    k, M    ph, k

Proof of Theorem modqmuladd
StepHypRef Expression
1 modqmuladd.a . . . . . . 7  |-  ( ph  ->  A  e.  ZZ )
2 zq 9655 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  QQ )
31, 2syl 14 . . . . . 6  |-  ( ph  ->  A  e.  QQ )
4 modqmuladd.m . . . . . 6  |-  ( ph  ->  M  e.  QQ )
5 modqmuladd.mgt0 . . . . . . 7  |-  ( ph  ->  0  <  M )
65gt0ne0d 8498 . . . . . 6  |-  ( ph  ->  M  =/=  0 )
7 qdivcl 9672 . . . . . 6  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  M  =/=  0 )  ->  ( A  /  M )  e.  QQ )
83, 4, 6, 7syl3anc 1249 . . . . 5  |-  ( ph  ->  ( A  /  M
)  e.  QQ )
98flqcld 10307 . . . 4  |-  ( ph  ->  ( |_ `  ( A  /  M ) )  e.  ZZ )
10 oveq1 5902 . . . . . . 7  |-  ( k  =  ( |_ `  ( A  /  M
) )  ->  (
k  x.  M )  =  ( ( |_
`  ( A  /  M ) )  x.  M ) )
1110oveq1d 5910 . . . . . 6  |-  ( k  =  ( |_ `  ( A  /  M
) )  ->  (
( k  x.  M
)  +  ( A  mod  M ) )  =  ( ( ( |_ `  ( A  /  M ) )  x.  M )  +  ( A  mod  M
) ) )
1211eqeq2d 2201 . . . . 5  |-  ( k  =  ( |_ `  ( A  /  M
) )  ->  ( A  =  ( (
k  x.  M )  +  ( A  mod  M ) )  <->  A  =  ( ( ( |_
`  ( A  /  M ) )  x.  M )  +  ( A  mod  M ) ) ) )
1312adantl 277 . . . 4  |-  ( (
ph  /\  k  =  ( |_ `  ( A  /  M ) ) )  ->  ( A  =  ( ( k  x.  M )  +  ( A  mod  M
) )  <->  A  =  ( ( ( |_
`  ( A  /  M ) )  x.  M )  +  ( A  mod  M ) ) ) )
14 flqpmodeq 10357 . . . . . 6  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( ( |_ `  ( A  /  M
) )  x.  M
)  +  ( A  mod  M ) )  =  A )
153, 4, 5, 14syl3anc 1249 . . . . 5  |-  ( ph  ->  ( ( ( |_
`  ( A  /  M ) )  x.  M )  +  ( A  mod  M ) )  =  A )
1615eqcomd 2195 . . . 4  |-  ( ph  ->  A  =  ( ( ( |_ `  ( A  /  M ) )  x.  M )  +  ( A  mod  M
) ) )
179, 13, 16rspcedvd 2862 . . 3  |-  ( ph  ->  E. k  e.  ZZ  A  =  ( (
k  x.  M )  +  ( A  mod  M ) ) )
18 oveq2 5903 . . . . . 6  |-  ( B  =  ( A  mod  M )  ->  ( (
k  x.  M )  +  B )  =  ( ( k  x.  M )  +  ( A  mod  M ) ) )
1918eqeq2d 2201 . . . . 5  |-  ( B  =  ( A  mod  M )  ->  ( A  =  ( ( k  x.  M )  +  B )  <->  A  =  ( ( k  x.  M )  +  ( A  mod  M ) ) ) )
2019eqcoms 2192 . . . 4  |-  ( ( A  mod  M )  =  B  ->  ( A  =  ( (
k  x.  M )  +  B )  <->  A  =  ( ( k  x.  M )  +  ( A  mod  M ) ) ) )
2120rexbidv 2491 . . 3  |-  ( ( A  mod  M )  =  B  ->  ( E. k  e.  ZZ  A  =  ( (
k  x.  M )  +  B )  <->  E. k  e.  ZZ  A  =  ( ( k  x.  M
)  +  ( A  mod  M ) ) ) )
2217, 21syl5ibrcom 157 . 2  |-  ( ph  ->  ( ( A  mod  M )  =  B  ->  E. k  e.  ZZ  A  =  ( (
k  x.  M )  +  B ) ) )
23 oveq1 5902 . . . . . 6  |-  ( A  =  ( ( k  x.  M )  +  B )  ->  ( A  mod  M )  =  ( ( ( k  x.  M )  +  B )  mod  M
) )
2423adantl 277 . . . . 5  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  -> 
( A  mod  M
)  =  ( ( ( k  x.  M
)  +  B )  mod  M ) )
25 simplr 528 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  -> 
k  e.  ZZ )
264ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  ->  M  e.  QQ )
27 modqmuladd.bq . . . . . . 7  |-  ( ph  ->  B  e.  QQ )
2827ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  ->  B  e.  QQ )
29 modqmuladd.b . . . . . . 7  |-  ( ph  ->  B  e.  ( 0 [,) M ) )
3029ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  ->  B  e.  ( 0 [,) M ) )
31 mulqaddmodid 10394 . . . . . 6  |-  ( ( ( k  e.  ZZ  /\  M  e.  QQ )  /\  ( B  e.  QQ  /\  B  e.  ( 0 [,) M
) ) )  -> 
( ( ( k  x.  M )  +  B )  mod  M
)  =  B )
3225, 26, 28, 30, 31syl22anc 1250 . . . . 5  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  -> 
( ( ( k  x.  M )  +  B )  mod  M
)  =  B )
3324, 32eqtrd 2222 . . . 4  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  -> 
( A  mod  M
)  =  B )
3433ex 115 . . 3  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( A  =  ( ( k  x.  M )  +  B )  ->  ( A  mod  M )  =  B ) )
3534rexlimdva 2607 . 2  |-  ( ph  ->  ( E. k  e.  ZZ  A  =  ( ( k  x.  M
)  +  B )  ->  ( A  mod  M )  =  B ) )
3622, 35impbid 129 1  |-  ( ph  ->  ( ( 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    = wceq 1364    e. wcel 2160    =/= wne 2360   E.wrex 2469   class class class wbr 4018   ` cfv 5235  (class class class)co 5895   0cc0 7840    + caddc 7843    x. cmul 7845    < clt 8021    / cdiv 8658   ZZcz 9282   QQcq 9648   [,)cico 9919   |_cfl 10298    mod cmo 10352
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-n0 9206  df-z 9283  df-q 9649  df-rp 9683  df-ico 9923  df-fl 10300  df-mod 10353
This theorem is referenced by:  modqmuladdim  10397
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