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Theorem modqadd1 10391
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 10354 . . . . . . 7  |-  ( ( A  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( A  mod  D )  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) ) )
62, 3, 4, 5syl3anc 1249 . . . . . 6  |-  ( ph  ->  ( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
7 modqadd1.b . . . . . . 7  |-  ( ph  ->  B  e.  QQ )
8 modqval 10354 . . . . . . 7  |-  ( ( B  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( B  mod  D )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) )
97, 3, 4, 8syl3anc 1249 . . . . . 6  |-  ( ph  ->  ( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
106, 9eqeq12d 2204 . . . . 5  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  <->  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) ) )
11 oveq1 5902 . . . . 5  |-  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  ->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C
)  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  +  C ) )
1210, 11biimtrdi 163 . . . 4  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  -> 
( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  +  C
) ) )
13 qcn 9663 . . . . . . 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 9663 . . . . . . 7  |-  ( C  e.  QQ  ->  C  e.  CC )
1715, 16syl 14 . . . . . 6  |-  ( ph  ->  C  e.  CC )
18 qcn 9663 . . . . . . . 8  |-  ( D  e.  QQ  ->  D  e.  CC )
193, 18syl 14 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
204gt0ne0d 8498 . . . . . . . . . 10  |-  ( ph  ->  D  =/=  0 )
21 qdivcl 9672 . . . . . . . . . 10  |-  ( ( A  e.  QQ  /\  D  e.  QQ  /\  D  =/=  0 )  ->  ( A  /  D )  e.  QQ )
222, 3, 20, 21syl3anc 1249 . . . . . . . . 9  |-  ( ph  ->  ( A  /  D
)  e.  QQ )
2322flqcld 10307 . . . . . . . 8  |-  ( ph  ->  ( |_ `  ( A  /  D ) )  e.  ZZ )
2423zcnd 9405 . . . . . . 7  |-  ( ph  ->  ( |_ `  ( A  /  D ) )  e.  CC )
2519, 24mulcld 8007 . . . . . 6  |-  ( ph  ->  ( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
2614, 17, 25addsubd 8318 . . . . 5  |-  ( ph  ->  ( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C
) )
27 qcn 9663 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  CC )
287, 27syl 14 . . . . . 6  |-  ( ph  ->  B  e.  CC )
29 qdivcl 9672 . . . . . . . . . 10  |-  ( ( B  e.  QQ  /\  D  e.  QQ  /\  D  =/=  0 )  ->  ( B  /  D )  e.  QQ )
307, 3, 20, 29syl3anc 1249 . . . . . . . . 9  |-  ( ph  ->  ( B  /  D
)  e.  QQ )
3130flqcld 10307 . . . . . . . 8  |-  ( ph  ->  ( |_ `  ( B  /  D ) )  e.  ZZ )
3231zcnd 9405 . . . . . . 7  |-  ( ph  ->  ( |_ `  ( B  /  D ) )  e.  CC )
3319, 32mulcld 8007 . . . . . 6  |-  ( ph  ->  ( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
3428, 17, 33addsubd 8318 . . . . 5  |-  ( ph  ->  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  +  C
) )
3526, 34eqeq12d 2204 . . . 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 169 . . 3  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  -> 
( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) ) )
37 oveq1 5902 . . . 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 9664 . . . . . . 7  |-  ( ( A  e.  QQ  /\  C  e.  QQ )  ->  ( A  +  C
)  e.  QQ )
392, 15, 38syl2anc 411 . . . . . 6  |-  ( ph  ->  ( A  +  C
)  e.  QQ )
40 modqcyc2 10390 . . . . . 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 1250 . . . . 5  |-  ( ph  ->  ( ( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  mod  D )  =  ( ( A  +  C )  mod 
D ) )
42 qaddcl 9664 . . . . . . 7  |-  ( ( B  e.  QQ  /\  C  e.  QQ )  ->  ( B  +  C
)  e.  QQ )
437, 15, 42syl2anc 411 . . . . . 6  |-  ( ph  ->  ( B  +  C
)  e.  QQ )
44 modqcyc2 10390 . . . . . 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 1250 . . . . 5  |-  ( ph  ->  ( ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  mod  D )  =  ( ( B  +  C )  mod 
D ) )
4641, 45eqeq12d 2204 . . . 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, 46imbitrid 154 . . 3  |-  ( ph  ->  ( ( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  ->  (
( A  +  C
)  mod  D )  =  ( ( B  +  C )  mod 
D ) ) )
4836, 47syld 45 . 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 1364    e. wcel 2160    =/= wne 2360   class class class wbr 4018   ` cfv 5235  (class class class)co 5895   CCcc 7838   0cc0 7840    + caddc 7843    x. cmul 7845    < clt 8021    - cmin 8157    / cdiv 8658   ZZcz 9282   QQcq 9648   |_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-fl 10300  df-mod 10353
This theorem is referenced by:  modqaddabs  10392  modqaddmod  10393  modqadd12d  10410  modqaddmulmod  10421  moddvds  11837  lgsvalmod  14873  lgsmod  14880  lgsne0  14892
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