MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  modadd1 Structured version   Unicode version

Theorem modadd1 11745
Description: Addition property of the modulo operation. (Contributed by NM, 12-Nov-2008.)
Assertion
Ref Expression
modadd1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A  +  C
)  mod  D )  =  ( ( B  +  C )  mod 
D ) )

Proof of Theorem modadd1
StepHypRef Expression
1 modval 11710 . . . . . . . 8  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
2 modval 11710 . . . . . . . 8  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
31, 2eqeqan12d 2458 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  D  e.  RR+ )  /\  ( B  e.  RR  /\  D  e.  RR+ )
)  ->  ( ( A  mod  D )  =  ( B  mod  D
)  <->  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) ) )
43anandirs 827 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  RR+ )  ->  ( ( A  mod  D )  =  ( B  mod  D
)  <->  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) ) )
54adantrl 715 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ ) )  ->  (
( A  mod  D
)  =  ( B  mod  D )  <->  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) ) ) )
6 oveq1 6098 . . . . 5  |-  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  ->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C
)  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  +  C ) )
75, 6syl6bi 228 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ ) )  ->  (
( A  mod  D
)  =  ( B  mod  D )  -> 
( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  +  C
) ) )
8 recn 9372 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
98adantr 465 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  A  e.  CC )
10 recn 9372 . . . . . . . 8  |-  ( C  e.  RR  ->  C  e.  CC )
1110ad2antrl 727 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  C  e.  CC )
12 rpcn 10999 . . . . . . . . . 10  |-  ( D  e.  RR+  ->  D  e.  CC )
1312adantl 466 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  ->  D  e.  CC )
14 rerpdivcl 11018 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  /  D
)  e.  RR )
15 reflcl 11646 . . . . . . . . . . 11  |-  ( ( A  /  D )  e.  RR  ->  ( |_ `  ( A  /  D ) )  e.  RR )
1615recnd 9412 . . . . . . . . . 10  |-  ( ( A  /  D )  e.  RR  ->  ( |_ `  ( A  /  D ) )  e.  CC )
1714, 16syl 16 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( A  /  D ) )  e.  CC )
1813, 17mulcld 9406 . . . . . . . 8  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
1918adantrl 715 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  ( D  x.  ( |_ `  ( A  /  D ) ) )  e.  CC )
209, 11, 19addsubd 9740 . . . . . 6  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  ( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  =  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  +  C ) )
2120adantlr 714 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ ) )  ->  (
( A  +  C
)  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C
) )
22 recn 9372 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  CC )
2322adantr 465 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  B  e.  CC )
2410ad2antrl 727 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  C  e.  CC )
2512adantl 466 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  ->  D  e.  CC )
26 rerpdivcl 11018 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  /  D
)  e.  RR )
27 reflcl 11646 . . . . . . . . . . 11  |-  ( ( B  /  D )  e.  RR  ->  ( |_ `  ( B  /  D ) )  e.  RR )
2827recnd 9412 . . . . . . . . . 10  |-  ( ( B  /  D )  e.  RR  ->  ( |_ `  ( B  /  D ) )  e.  CC )
2926, 28syl 16 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( B  /  D ) )  e.  CC )
3025, 29mulcld 9406 . . . . . . . 8  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
3130adantrl 715 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  ( D  x.  ( |_ `  ( B  /  D ) ) )  e.  CC )
3223, 24, 31addsubd 9740 . . . . . 6  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  +  C ) )
3332adantll 713 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ ) )  ->  (
( B  +  C
)  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  +  C
) )
3421, 33eqeq12d 2457 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ ) )  ->  (
( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  <->  ( ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  +  C
)  =  ( ( B  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  +  C ) ) )
357, 34sylibrd 234 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ ) )  ->  (
( A  mod  D
)  =  ( B  mod  D )  -> 
( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) ) )
36 oveq1 6098 . . . 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 ) )
37 readdcl 9365 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  +  C
)  e.  RR )
3837adantrr 716 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  ( A  +  C )  e.  RR )
39 simprr 756 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  D  e.  RR+ )
4014flcld 11648 . . . . . . . 8  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( A  /  D ) )  e.  ZZ )
4140adantrl 715 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  ( |_ `  ( A  /  D
) )  e.  ZZ )
42 modcyc2 11744 . . . . . . 7  |-  ( ( ( A  +  C
)  e.  RR  /\  D  e.  RR+  /\  ( |_ `  ( A  /  D ) )  e.  ZZ )  ->  (
( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  mod  D )  =  ( ( A  +  C )  mod  D
) )
4338, 39, 41, 42syl3anc 1218 . . . . . 6  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  ( (
( A  +  C
)  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  mod  D )  =  ( ( A  +  C )  mod  D
) )
4443adantlr 714 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ ) )  ->  (
( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  mod  D )  =  ( ( A  +  C )  mod  D
) )
45 readdcl 9365 . . . . . . . 8  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  +  C
)  e.  RR )
4645adantrr 716 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  ( B  +  C )  e.  RR )
47 simprr 756 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  D  e.  RR+ )
4826flcld 11648 . . . . . . . 8  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( B  /  D ) )  e.  ZZ )
4948adantrl 715 . . . . . . 7  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  ( |_ `  ( B  /  D
) )  e.  ZZ )
50 modcyc2 11744 . . . . . . 7  |-  ( ( ( B  +  C
)  e.  RR  /\  D  e.  RR+  /\  ( |_ `  ( B  /  D ) )  e.  ZZ )  ->  (
( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  mod  D )  =  ( ( B  +  C )  mod  D
) )
5146, 47, 49, 50syl3anc 1218 . . . . . 6  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  ( (
( B  +  C
)  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  mod  D )  =  ( ( B  +  C )  mod  D
) )
5251adantll 713 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ ) )  ->  (
( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D
) ) ) )  mod  D )  =  ( ( B  +  C )  mod  D
) )
5344, 52eqeq12d 2457 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ ) )  ->  (
( ( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) )  mod  D )  =  ( ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  mod  D
)  <->  ( ( A  +  C )  mod 
D )  =  ( ( B  +  C
)  mod  D )
) )
5436, 53syl5ib 219 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ ) )  ->  (
( ( A  +  C )  -  ( D  x.  ( |_ `  ( A  /  D
) ) ) )  =  ( ( B  +  C )  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) )  ->  ( ( A  +  C )  mod  D )  =  ( ( B  +  C
)  mod  D )
) )
5535, 54syld 44 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ ) )  ->  (
( A  mod  D
)  =  ( B  mod  D )  -> 
( ( A  +  C )  mod  D
)  =  ( ( B  +  C )  mod  D ) ) )
56553impia 1184 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A  +  C
)  mod  D )  =  ( ( B  +  C )  mod 
D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281    + caddc 9285    x. cmul 9287    - cmin 9595    / cdiv 9993   ZZcz 10646   RR+crp 10991   |_cfl 11640    mod cmo 11708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fl 11642  df-mod 11709
This theorem is referenced by:  modaddabs  11746  modaddmod  11747  modadd12d  11755  modaddmulmod  11765  moddvds  13542  modsubi  14101  lgslem4  22638  lgsvalmod  22654  lgsmod  22660  lgsne0  22672  lgseisen  22692  pellexlem6  29175
  Copyright terms: Public domain W3C validator