Users' Mathboxes Mathbox for Paul Chapman < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  modaddabs Unicode version

Theorem modaddabs 24026
Description: Absorbtion law for modulo. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
modaddabs  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( A  mod  C )  +  ( B  mod  C ) )  mod  C )  =  ( ( A  +  B )  mod  C
) )

Proof of Theorem modaddabs
StepHypRef Expression
1 modcl 10992 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( A  mod  C
)  e.  RR )
21recnd 8877 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( A  mod  C
)  e.  CC )
323adant2 974 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( A  mod  C )  e.  CC )
4 modcl 10992 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( B  mod  C
)  e.  RR )
54recnd 8877 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( B  mod  C
)  e.  CC )
653adant1 973 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( B  mod  C )  e.  CC )
73, 6addcomd 9030 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  +  ( B  mod  C ) )  =  ( ( B  mod  C )  +  ( A  mod  C
) ) )
87oveq1d 5889 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( A  mod  C )  +  ( B  mod  C ) )  mod  C )  =  ( ( ( B  mod  C )  +  ( A  mod  C
) )  mod  C
) )
9 simpl 443 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  ->  B  e.  RR )
104, 9jca 518 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( ( B  mod  C )  e.  RR  /\  B  e.  RR )
)
11103adant1 973 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( B  mod  C
)  e.  RR  /\  B  e.  RR )
)
12 simpr 447 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  ->  C  e.  RR+ )
131, 12jca 518 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( ( A  mod  C )  e.  RR  /\  C  e.  RR+ ) )
14133adant2 974 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  e.  RR  /\  C  e.  RR+ ) )
15 modabs2 11014 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( ( B  mod  C )  mod  C )  =  ( B  mod  C ) )
16153adant1 973 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( B  mod  C
)  mod  C )  =  ( B  mod  C ) )
17 modadd1 11017 . . . 4  |-  ( ( ( ( B  mod  C )  e.  RR  /\  B  e.  RR )  /\  ( ( A  mod  C )  e.  RR  /\  C  e.  RR+ )  /\  ( ( B  mod  C )  mod  C )  =  ( B  mod  C ) )  ->  (
( ( B  mod  C )  +  ( A  mod  C ) )  mod  C )  =  ( ( B  +  ( A  mod  C ) )  mod  C ) )
1811, 14, 16, 17syl3anc 1182 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( B  mod  C )  +  ( A  mod  C ) )  mod  C )  =  ( ( B  +  ( A  mod  C ) )  mod  C ) )
19 recn 8843 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
20193ad2ant2 977 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  B  e.  CC )
213, 20addcomd 9030 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  +  B )  =  ( B  +  ( A  mod  C ) ) )
2221oveq1d 5889 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( A  mod  C )  +  B )  mod  C )  =  ( ( B  +  ( A  mod  C ) )  mod  C ) )
2318, 22eqtr4d 2331 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( B  mod  C )  +  ( A  mod  C ) )  mod  C )  =  ( ( ( A  mod  C )  +  B )  mod  C
) )
24 simpl 443 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  ->  A  e.  RR )
251, 24jca 518 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( ( A  mod  C )  e.  RR  /\  A  e.  RR )
)
26253adant2 974 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  e.  RR  /\  A  e.  RR )
)
27 3simpc 954 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( B  e.  RR  /\  C  e.  RR+ ) )
28 modabs2 11014 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( ( A  mod  C )  mod  C )  =  ( A  mod  C ) )
29283adant2 974 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  mod  C )  =  ( A  mod  C ) )
30 modadd1 11017 . . 3  |-  ( ( ( ( A  mod  C )  e.  RR  /\  A  e.  RR )  /\  ( B  e.  RR  /\  C  e.  RR+ )  /\  ( ( A  mod  C )  mod  C )  =  ( A  mod  C ) )  ->  (
( ( A  mod  C )  +  B )  mod  C )  =  ( ( A  +  B )  mod  C
) )
3126, 27, 29, 30syl3anc 1182 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( A  mod  C )  +  B )  mod  C )  =  ( ( A  +  B )  mod  C
) )
328, 23, 313eqtrd 2332 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( A  mod  C )  +  ( B  mod  C ) )  mod  C )  =  ( ( A  +  B )  mod  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751   RRcr 8752    + caddc 8756   RR+crp 10370    mod cmo 10989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-mod 10990
  Copyright terms: Public domain W3C validator