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Theorem modqmuladd 10032
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 9320 . . . . . . 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 8193 . . . . . 6  |-  ( ph  ->  M  =/=  0 )
7 qdivcl 9337 . . . . . 6  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  M  =/=  0 )  ->  ( A  /  M )  e.  QQ )
83, 4, 6, 7syl3anc 1199 . . . . 5  |-  ( ph  ->  ( A  /  M
)  e.  QQ )
98flqcld 9943 . . . 4  |-  ( ph  ->  ( |_ `  ( A  /  M ) )  e.  ZZ )
10 oveq1 5735 . . . . . . 7  |-  ( k  =  ( |_ `  ( A  /  M
) )  ->  (
k  x.  M )  =  ( ( |_
`  ( A  /  M ) )  x.  M ) )
1110oveq1d 5743 . . . . . 6  |-  ( k  =  ( |_ `  ( A  /  M
) )  ->  (
( k  x.  M
)  +  ( A  mod  M ) )  =  ( ( ( |_ `  ( A  /  M ) )  x.  M )  +  ( A  mod  M
) ) )
1211eqeq2d 2126 . . . . 5  |-  ( k  =  ( |_ `  ( A  /  M
) )  ->  ( A  =  ( (
k  x.  M )  +  ( A  mod  M ) )  <->  A  =  ( ( ( |_
`  ( A  /  M ) )  x.  M )  +  ( A  mod  M ) ) ) )
1312adantl 273 . . . 4  |-  ( (
ph  /\  k  =  ( |_ `  ( A  /  M ) ) )  ->  ( A  =  ( ( k  x.  M )  +  ( A  mod  M
) )  <->  A  =  ( ( ( |_
`  ( A  /  M ) )  x.  M )  +  ( A  mod  M ) ) ) )
14 flqpmodeq 9993 . . . . . 6  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( ( |_ `  ( A  /  M
) )  x.  M
)  +  ( A  mod  M ) )  =  A )
153, 4, 5, 14syl3anc 1199 . . . . 5  |-  ( ph  ->  ( ( ( |_
`  ( A  /  M ) )  x.  M )  +  ( A  mod  M ) )  =  A )
1615eqcomd 2120 . . . 4  |-  ( ph  ->  A  =  ( ( ( |_ `  ( A  /  M ) )  x.  M )  +  ( A  mod  M
) ) )
179, 13, 16rspcedvd 2766 . . 3  |-  ( ph  ->  E. k  e.  ZZ  A  =  ( (
k  x.  M )  +  ( A  mod  M ) ) )
18 oveq2 5736 . . . . . 6  |-  ( B  =  ( A  mod  M )  ->  ( (
k  x.  M )  +  B )  =  ( ( k  x.  M )  +  ( A  mod  M ) ) )
1918eqeq2d 2126 . . . . 5  |-  ( B  =  ( A  mod  M )  ->  ( A  =  ( ( k  x.  M )  +  B )  <->  A  =  ( ( k  x.  M )  +  ( A  mod  M ) ) ) )
2019eqcoms 2118 . . . 4  |-  ( ( A  mod  M )  =  B  ->  ( A  =  ( (
k  x.  M )  +  B )  <->  A  =  ( ( k  x.  M )  +  ( A  mod  M ) ) ) )
2120rexbidv 2412 . . 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 156 . 2  |-  ( ph  ->  ( ( A  mod  M )  =  B  ->  E. k  e.  ZZ  A  =  ( (
k  x.  M )  +  B ) ) )
23 oveq1 5735 . . . . . 6  |-  ( A  =  ( ( k  x.  M )  +  B )  ->  ( A  mod  M )  =  ( ( ( k  x.  M )  +  B )  mod  M
) )
2423adantl 273 . . . . 5  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  -> 
( A  mod  M
)  =  ( ( ( k  x.  M
)  +  B )  mod  M ) )
25 simplr 502 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  -> 
k  e.  ZZ )
264ad2antrr 477 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  ->  M  e.  QQ )
27 modqmuladd.bq . . . . . . 7  |-  ( ph  ->  B  e.  QQ )
2827ad2antrr 477 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  ->  B  e.  QQ )
29 modqmuladd.b . . . . . . 7  |-  ( ph  ->  B  e.  ( 0 [,) M ) )
3029ad2antrr 477 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  ->  B  e.  ( 0 [,) M ) )
31 mulqaddmodid 10030 . . . . . 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 1200 . . . . 5  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  -> 
( ( ( k  x.  M )  +  B )  mod  M
)  =  B )
3324, 32eqtrd 2147 . . . 4  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  -> 
( A  mod  M
)  =  B )
3433ex 114 . . 3  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( A  =  ( ( k  x.  M )  +  B )  ->  ( A  mod  M )  =  B ) )
3534rexlimdva 2523 . 2  |-  ( ph  ->  ( E. k  e.  ZZ  A  =  ( ( k  x.  M
)  +  B )  ->  ( A  mod  M )  =  B ) )
3622, 35impbid 128 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 103    <-> wb 104    = wceq 1314    e. wcel 1463    =/= wne 2282   E.wrex 2391   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   0cc0 7547    + caddc 7550    x. cmul 7552    < clt 7724    / cdiv 8345   ZZcz 8958   QQcq 9313   [,)cico 9566   |_cfl 9934    mod cmo 9988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-po 4178  df-iso 4179  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-n0 8882  df-z 8959  df-q 9314  df-rp 9344  df-ico 9570  df-fl 9936  df-mod 9989
This theorem is referenced by:  modqmuladdim  10033
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