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Theorem modaddabs 24894
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 11180 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( A  mod  C
)  e.  RR )
21recnd 9047 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( A  mod  C
)  e.  CC )
323adant2 976 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( A  mod  C )  e.  CC )
4 modcl 11180 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( B  mod  C
)  e.  RR )
54recnd 9047 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( B  mod  C
)  e.  CC )
653adant1 975 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( B  mod  C )  e.  CC )
73, 6addcomd 9200 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  +  ( B  mod  C ) )  =  ( ( B  mod  C )  +  ( A  mod  C
) ) )
87oveq1d 6035 . 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 444 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  ->  B  e.  RR )
104, 9jca 519 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( ( B  mod  C )  e.  RR  /\  B  e.  RR )
)
11103adant1 975 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( B  mod  C
)  e.  RR  /\  B  e.  RR )
)
12 simpr 448 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  ->  C  e.  RR+ )
131, 12jca 519 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( ( A  mod  C )  e.  RR  /\  C  e.  RR+ ) )
14133adant2 976 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  e.  RR  /\  C  e.  RR+ ) )
15 modabs2 11202 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( ( B  mod  C )  mod  C )  =  ( B  mod  C ) )
16153adant1 975 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( B  mod  C
)  mod  C )  =  ( B  mod  C ) )
17 modadd1 11205 . . . 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 1184 . . 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 9013 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
20193ad2ant2 979 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  B  e.  CC )
213, 20addcomd 9200 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  +  B )  =  ( B  +  ( A  mod  C ) ) )
2221oveq1d 6035 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( A  mod  C )  +  B )  mod  C )  =  ( ( B  +  ( A  mod  C ) )  mod  C ) )
2318, 22eqtr4d 2422 . 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 444 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  ->  A  e.  RR )
251, 24jca 519 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( ( A  mod  C )  e.  RR  /\  A  e.  RR )
)
26253adant2 976 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  e.  RR  /\  A  e.  RR )
)
27 3simpc 956 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( B  e.  RR  /\  C  e.  RR+ ) )
28 modabs2 11202 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( ( A  mod  C )  mod  C )  =  ( A  mod  C ) )
29283adant2 976 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  mod  C )  =  ( A  mod  C ) )
30 modadd1 11205 . . 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 1184 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( A  mod  C )  +  B )  mod  C )  =  ( ( A  +  B )  mod  C
) )
328, 23, 313eqtrd 2423 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 359    /\ w3a 936    = wceq 1649    e. wcel 1717  (class class class)co 6020   CCcc 8921   RRcr 8922    + caddc 8926   RR+crp 10544    mod cmo 11177
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fl 11129  df-mod 11178
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