ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  modqadd1 Unicode version

Theorem modqadd1 9764
Description: Addition property of the modulo operation. (Contributed by Jim Kingdon, 22-Oct-2021.)
Hypotheses
Ref Expression
modqadd1.a  |-  ( ph  ->  A  e.  QQ )
modqadd1.b  |-  ( ph  ->  B  e.  QQ )
modqadd1.c  |-  ( ph  ->  C  e.  QQ )
modqadd1.dq  |-  ( ph  ->  D  e.  QQ )
modqadd1.dgt0  |-  ( ph  ->  0  <  D )
modqadd1.ab  |-  ( ph  ->  ( A  mod  D
)  =  ( B  mod  D ) )
Assertion
Ref Expression
modqadd1  |-  ( ph  ->  ( ( A  +  C )  mod  D
)  =  ( ( B  +  C )  mod  D ) )

Proof of Theorem modqadd1
StepHypRef Expression
1 modqadd1.ab . 2  |-  ( ph  ->  ( A  mod  D
)  =  ( B  mod  D ) )
2 modqadd1.a . . . . . . 7  |-  ( ph  ->  A  e.  QQ )
3 modqadd1.dq . . . . . . 7  |-  ( ph  ->  D  e.  QQ )
4 modqadd1.dgt0 . . . . . . 7  |-  ( ph  ->  0  <  D )
5 modqval 9727 . . . . . . 7  |-  ( ( A  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( A  mod  D )  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) ) )
62, 3, 4, 5syl3anc 1174 . . . . . 6  |-  ( ph  ->  ( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
7 modqadd1.b . . . . . . 7  |-  ( ph  ->  B  e.  QQ )
8 modqval 9727 . . . . . . 7  |-  ( ( B  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( B  mod  D )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) )
97, 3, 4, 8syl3anc 1174 . . . . . 6  |-  ( ph  ->  ( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
106, 9eqeq12d 2102 . . . . 5  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  <->  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) ) )
11 oveq1 5659 . . . . 5  |-  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  ->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C
)  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  +  C ) )
1210, 11syl6bi 161 . . . 4  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  -> 
( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  +  C
) ) )
13 qcn 9117 . . . . . . 7  |-  ( A  e.  QQ  ->  A  e.  CC )
142, 13syl 14 . . . . . 6  |-  ( ph  ->  A  e.  CC )
15 modqadd1.c . . . . . . 7  |-  ( ph  ->  C  e.  QQ )
16 qcn 9117 . . . . . . 7  |-  ( C  e.  QQ  ->  C  e.  CC )
1715, 16syl 14 . . . . . 6  |-  ( ph  ->  C  e.  CC )
18 qcn 9117 . . . . . . . 8  |-  ( D  e.  QQ  ->  D  e.  CC )
193, 18syl 14 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
204gt0ne0d 7988 . . . . . . . . . 10  |-  ( ph  ->  D  =/=  0 )
21 qdivcl 9126 . . . . . . . . . 10  |-  ( ( A  e.  QQ  /\  D  e.  QQ  /\  D  =/=  0 )  ->  ( A  /  D )  e.  QQ )
222, 3, 20, 21syl3anc 1174 . . . . . . . . 9  |-  ( ph  ->  ( A  /  D
)  e.  QQ )
2322flqcld 9680 . . . . . . . 8  |-  ( ph  ->  ( |_ `  ( A  /  D ) )  e.  ZZ )
2423zcnd 8867 . . . . . . 7  |-  ( ph  ->  ( |_ `  ( A  /  D ) )  e.  CC )
2519, 24mulcld 7506 . . . . . 6  |-  ( ph  ->  ( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
2614, 17, 25addsubd 7812 . . . . 5  |-  ( ph  ->  ( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C
) )
27 qcn 9117 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  CC )
287, 27syl 14 . . . . . 6  |-  ( ph  ->  B  e.  CC )
29 qdivcl 9126 . . . . . . . . . 10  |-  ( ( B  e.  QQ  /\  D  e.  QQ  /\  D  =/=  0 )  ->  ( B  /  D )  e.  QQ )
307, 3, 20, 29syl3anc 1174 . . . . . . . . 9  |-  ( ph  ->  ( B  /  D
)  e.  QQ )
3130flqcld 9680 . . . . . . . 8  |-  ( ph  ->  ( |_ `  ( B  /  D ) )  e.  ZZ )
3231zcnd 8867 . . . . . . 7  |-  ( ph  ->  ( |_ `  ( B  /  D ) )  e.  CC )
3319, 32mulcld 7506 . . . . . 6  |-  ( ph  ->  ( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
3428, 17, 33addsubd 7812 . . . . 5  |-  ( ph  ->  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  +  C
) )
3526, 34eqeq12d 2102 . . . 4  |-  ( ph  ->  ( ( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  <->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C
)  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  +  C ) ) )
3612, 35sylibrd 167 . . 3  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  -> 
( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) ) )
37 oveq1 5659 . . . 4  |-  ( ( ( A  +  C
)  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  ->  ( (
( A  +  C
)  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  mod  D )  =  ( ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  mod  D ) )
38 qaddcl 9118 . . . . . . 7  |-  ( ( A  e.  QQ  /\  C  e.  QQ )  ->  ( A  +  C
)  e.  QQ )
392, 15, 38syl2anc 403 . . . . . 6  |-  ( ph  ->  ( A  +  C
)  e.  QQ )
40 modqcyc2 9763 . . . . . 6  |-  ( ( ( ( A  +  C )  e.  QQ  /\  ( |_ `  ( A  /  D ) )  e.  ZZ )  /\  ( D  e.  QQ  /\  0  <  D ) )  ->  ( (
( A  +  C
)  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  mod  D )  =  ( ( A  +  C )  mod  D
) )
4139, 23, 3, 4, 40syl22anc 1175 . . . . 5  |-  ( ph  ->  ( ( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  mod  D )  =  ( ( A  +  C )  mod 
D ) )
42 qaddcl 9118 . . . . . . 7  |-  ( ( B  e.  QQ  /\  C  e.  QQ )  ->  ( B  +  C
)  e.  QQ )
437, 15, 42syl2anc 403 . . . . . 6  |-  ( ph  ->  ( B  +  C
)  e.  QQ )
44 modqcyc2 9763 . . . . . 6  |-  ( ( ( ( B  +  C )  e.  QQ  /\  ( |_ `  ( B  /  D ) )  e.  ZZ )  /\  ( D  e.  QQ  /\  0  <  D ) )  ->  ( (
( B  +  C
)  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  mod  D )  =  ( ( B  +  C )  mod  D
) )
4543, 31, 3, 4, 44syl22anc 1175 . . . . 5  |-  ( ph  ->  ( ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  mod  D )  =  ( ( B  +  C )  mod 
D ) )
4641, 45eqeq12d 2102 . . . 4  |-  ( ph  ->  ( ( ( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  mod  D
)  =  ( ( ( B  +  C
)  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  mod  D )  <->  ( ( A  +  C )  mod  D )  =  ( ( B  +  C
)  mod  D )
) )
4737, 46syl5ib 152 . . 3  |-  ( ph  ->  ( ( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  ->  (
( A  +  C
)  mod  D )  =  ( ( B  +  C )  mod 
D ) ) )
4836, 47syld 44 . 2  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  -> 
( ( A  +  C )  mod  D
)  =  ( ( B  +  C )  mod  D ) ) )
491, 48mpd 13 1  |-  ( ph  ->  ( ( A  +  C )  mod  D
)  =  ( ( B  +  C )  mod  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438    =/= wne 2255   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   CCcc 7346   0cc0 7348    + caddc 7351    x. cmul 7353    < clt 7520    - cmin 7651    / cdiv 8137   ZZcz 8748   QQcq 9102   |_cfl 9671    mod cmo 9725
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 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462
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 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-po 4123  df-iso 4124  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-n0 8672  df-z 8749  df-q 9103  df-rp 9133  df-fl 9673  df-mod 9726
This theorem is referenced by:  modqaddabs  9765  modqaddmod  9766  modqadd12d  9783  modqaddmulmod  9794  moddvds  11079
  Copyright terms: Public domain W3C validator