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Theorem modqmuladd 10352
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 9615 . . . . . . 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 8459 . . . . . 6  |-  ( ph  ->  M  =/=  0 )
7 qdivcl 9632 . . . . . 6  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  M  =/=  0 )  ->  ( A  /  M )  e.  QQ )
83, 4, 6, 7syl3anc 1238 . . . . 5  |-  ( ph  ->  ( A  /  M
)  e.  QQ )
98flqcld 10263 . . . 4  |-  ( ph  ->  ( |_ `  ( A  /  M ) )  e.  ZZ )
10 oveq1 5876 . . . . . . 7  |-  ( k  =  ( |_ `  ( A  /  M
) )  ->  (
k  x.  M )  =  ( ( |_
`  ( A  /  M ) )  x.  M ) )
1110oveq1d 5884 . . . . . 6  |-  ( k  =  ( |_ `  ( A  /  M
) )  ->  (
( k  x.  M
)  +  ( A  mod  M ) )  =  ( ( ( |_ `  ( A  /  M ) )  x.  M )  +  ( A  mod  M
) ) )
1211eqeq2d 2189 . . . . 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 10313 . . . . . 6  |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M )  ->  (
( ( |_ `  ( A  /  M
) )  x.  M
)  +  ( A  mod  M ) )  =  A )
153, 4, 5, 14syl3anc 1238 . . . . 5  |-  ( ph  ->  ( ( ( |_
`  ( A  /  M ) )  x.  M )  +  ( A  mod  M ) )  =  A )
1615eqcomd 2183 . . . 4  |-  ( ph  ->  A  =  ( ( ( |_ `  ( A  /  M ) )  x.  M )  +  ( A  mod  M
) ) )
179, 13, 16rspcedvd 2847 . . 3  |-  ( ph  ->  E. k  e.  ZZ  A  =  ( (
k  x.  M )  +  ( A  mod  M ) ) )
18 oveq2 5877 . . . . . 6  |-  ( B  =  ( A  mod  M )  ->  ( (
k  x.  M )  +  B )  =  ( ( k  x.  M )  +  ( A  mod  M ) ) )
1918eqeq2d 2189 . . . . 5  |-  ( B  =  ( A  mod  M )  ->  ( A  =  ( ( k  x.  M )  +  B )  <->  A  =  ( ( k  x.  M )  +  ( A  mod  M ) ) ) )
2019eqcoms 2180 . . . 4  |-  ( ( A  mod  M )  =  B  ->  ( A  =  ( (
k  x.  M )  +  B )  <->  A  =  ( ( k  x.  M )  +  ( A  mod  M ) ) ) )
2120rexbidv 2478 . . 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 5876 . . . . . 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 10350 . . . . . 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 1239 . . . . 5  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  A  =  ( ( k  x.  M )  +  B ) )  -> 
( ( ( k  x.  M )  +  B )  mod  M
)  =  B )
3324, 32eqtrd 2210 . . . 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 2594 . 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 1353    e. wcel 2148    =/= wne 2347   E.wrex 2456   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   0cc0 7802    + caddc 7805    x. cmul 7807    < clt 7982    / cdiv 8618   ZZcz 9242   QQcq 9608   [,)cico 9877   |_cfl 10254    mod cmo 10308
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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-n0 9166  df-z 9243  df-q 9609  df-rp 9641  df-ico 9881  df-fl 10256  df-mod 10309
This theorem is referenced by:  modqmuladdim  10353
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