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Theorem modqadd1 10260
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 10223 . . . . . . 7  |-  ( ( A  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( A  mod  D )  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) ) )
62, 3, 4, 5syl3anc 1220 . . . . . 6  |-  ( ph  ->  ( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
7 modqadd1.b . . . . . . 7  |-  ( ph  ->  B  e.  QQ )
8 modqval 10223 . . . . . . 7  |-  ( ( B  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( B  mod  D )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) )
97, 3, 4, 8syl3anc 1220 . . . . . 6  |-  ( ph  ->  ( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
106, 9eqeq12d 2172 . . . . 5  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  <->  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) ) )
11 oveq1 5831 . . . . 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 162 . . . 4  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  -> 
( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  +  C
) ) )
13 qcn 9543 . . . . . . 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 9543 . . . . . . 7  |-  ( C  e.  QQ  ->  C  e.  CC )
1715, 16syl 14 . . . . . 6  |-  ( ph  ->  C  e.  CC )
18 qcn 9543 . . . . . . . 8  |-  ( D  e.  QQ  ->  D  e.  CC )
193, 18syl 14 . . . . . . 7  |-  ( ph  ->  D  e.  CC )
204gt0ne0d 8387 . . . . . . . . . 10  |-  ( ph  ->  D  =/=  0 )
21 qdivcl 9552 . . . . . . . . . 10  |-  ( ( A  e.  QQ  /\  D  e.  QQ  /\  D  =/=  0 )  ->  ( A  /  D )  e.  QQ )
222, 3, 20, 21syl3anc 1220 . . . . . . . . 9  |-  ( ph  ->  ( A  /  D
)  e.  QQ )
2322flqcld 10176 . . . . . . . 8  |-  ( ph  ->  ( |_ `  ( A  /  D ) )  e.  ZZ )
2423zcnd 9287 . . . . . . 7  |-  ( ph  ->  ( |_ `  ( A  /  D ) )  e.  CC )
2519, 24mulcld 7898 . . . . . 6  |-  ( ph  ->  ( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
2614, 17, 25addsubd 8207 . . . . 5  |-  ( ph  ->  ( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C
) )
27 qcn 9543 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  CC )
287, 27syl 14 . . . . . 6  |-  ( ph  ->  B  e.  CC )
29 qdivcl 9552 . . . . . . . . . 10  |-  ( ( B  e.  QQ  /\  D  e.  QQ  /\  D  =/=  0 )  ->  ( B  /  D )  e.  QQ )
307, 3, 20, 29syl3anc 1220 . . . . . . . . 9  |-  ( ph  ->  ( B  /  D
)  e.  QQ )
3130flqcld 10176 . . . . . . . 8  |-  ( ph  ->  ( |_ `  ( B  /  D ) )  e.  ZZ )
3231zcnd 9287 . . . . . . 7  |-  ( ph  ->  ( |_ `  ( B  /  D ) )  e.  CC )
3319, 32mulcld 7898 . . . . . 6  |-  ( ph  ->  ( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
3428, 17, 33addsubd 8207 . . . . 5  |-  ( ph  ->  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  +  C
) )
3526, 34eqeq12d 2172 . . . 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 168 . . 3  |-  ( ph  ->  ( ( A  mod  D )  =  ( B  mod  D )  -> 
( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) ) )
37 oveq1 5831 . . . 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 9544 . . . . . . 7  |-  ( ( A  e.  QQ  /\  C  e.  QQ )  ->  ( A  +  C
)  e.  QQ )
392, 15, 38syl2anc 409 . . . . . 6  |-  ( ph  ->  ( A  +  C
)  e.  QQ )
40 modqcyc2 10259 . . . . . 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 1221 . . . . 5  |-  ( ph  ->  ( ( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  mod  D )  =  ( ( A  +  C )  mod 
D ) )
42 qaddcl 9544 . . . . . . 7  |-  ( ( B  e.  QQ  /\  C  e.  QQ )  ->  ( B  +  C
)  e.  QQ )
437, 15, 42syl2anc 409 . . . . . 6  |-  ( ph  ->  ( B  +  C
)  e.  QQ )
44 modqcyc2 10259 . . . . . 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 1221 . . . . 5  |-  ( ph  ->  ( ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  mod  D )  =  ( ( B  +  C )  mod 
D ) )
4641, 45eqeq12d 2172 . . . 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 153 . . 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 1335    e. wcel 2128    =/= wne 2327   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   CCcc 7730   0cc0 7732    + caddc 7735    x. cmul 7737    < clt 7912    - cmin 8046    / cdiv 8545   ZZcz 9167   QQcq 9528   |_cfl 10167    mod cmo 10221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-po 4256  df-iso 4257  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-n0 9091  df-z 9168  df-q 9529  df-rp 9561  df-fl 10169  df-mod 10222
This theorem is referenced by:  modqaddabs  10261  modqaddmod  10262  modqadd12d  10279  modqaddmulmod  10290  moddvds  11695
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