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Theorem modaddabs 24011
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 10976 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( A  mod  C
)  e.  RR )
21recnd 8861 . . . . 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 10976 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( B  mod  C
)  e.  RR )
54recnd 8861 . . . . 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 9014 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  +  ( B  mod  C ) )  =  ( ( B  mod  C )  +  ( A  mod  C
) ) )
87oveq1d 5873 . 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 10998 . . . . 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 11001 . . . 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 8827 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
20193ad2ant2 977 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  B  e.  CC )
213, 20addcomd 9014 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  +  B )  =  ( B  +  ( A  mod  C ) ) )
2221oveq1d 5873 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( A  mod  C )  +  B )  mod  C )  =  ( ( B  +  ( A  mod  C ) )  mod  C ) )
2318, 22eqtr4d 2318 . 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 10998 . . . 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 11001 . . 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 2319 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 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   RRcr 8736    + caddc 8740   RR+crp 10354    mod cmo 10973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974
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