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Theorem modadd1 12002
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 11967 . . . . . . . 8  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  mod  D
)  =  ( A  -  ( D  x.  ( |_ `  ( A  /  D ) ) ) ) )
2 modval 11967 . . . . . . . 8  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  mod  D
)  =  ( B  -  ( D  x.  ( |_ `  ( B  /  D ) ) ) ) )
31, 2eqeqan12d 2490 . . . . . . 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 829 . . . . . 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 6292 . . . . 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 9583 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
98adantr 465 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  A  e.  CC )
10 recn 9583 . . . . . . . 8  |-  ( C  e.  RR  ->  C  e.  CC )
1110ad2antrl 727 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  D  e.  RR+ )
)  ->  C  e.  CC )
12 rpcn 11229 . . . . . . . . . 10  |-  ( D  e.  RR+  ->  D  e.  CC )
1312adantl 466 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  ->  D  e.  CC )
14 rerpdivcl 11248 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  D  e.  RR+ )  -> 
( A  /  D
)  e.  RR )
15 reflcl 11902 . . . . . . . . . . 11  |-  ( ( A  /  D )  e.  RR  ->  ( |_ `  ( A  /  D ) )  e.  RR )
1615recnd 9623 . . . . . . . . . 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 9617 . . . . . . . 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 9952 . . . . . 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 9583 . . . . . . . 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 11248 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  D  e.  RR+ )  -> 
( B  /  D
)  e.  RR )
27 reflcl 11902 . . . . . . . . . . 11  |-  ( ( B  /  D )  e.  RR  ->  ( |_ `  ( B  /  D ) )  e.  RR )
2827recnd 9623 . . . . . . . . . 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 9617 . . . . . . . 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 9952 . . . . . 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 2489 . . . 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 6292 . . . 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 9576 . . . . . . . 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 11904 . . . . . . . 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 12001 . . . . . . 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 1228 . . . . . 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 9576 . . . . . . . 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 11904 . . . . . . . 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 12001 . . . . . . 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 1228 . . . . . 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 2489 . . . 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 1193 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 973    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6285   CCcc 9491   RRcr 9492    + caddc 9496    x. cmul 9498    - cmin 9806    / cdiv 10207   ZZcz 10865   RR+crp 11221   |_cfl 11896    mod cmo 11965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fl 11898  df-mod 11966
This theorem is referenced by:  modaddabs  12003  modaddmod  12004  modadd12d  12012  modaddmulmod  12022  moddvds  13857  modsubi  14420  lgslem4  23399  lgsvalmod  23415  lgsmod  23421  lgsne0  23433  lgseisen  23453  pellexlem6  30601
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