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Theorem modaddabs 25107
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 11245 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( A  mod  C
)  e.  RR )
21recnd 9106 . . . . 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 11245 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( B  mod  C
)  e.  RR )
54recnd 9106 . . . . 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 9260 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  +  ( B  mod  C ) )  =  ( ( B  mod  C )  +  ( A  mod  C
) ) )
87oveq1d 6088 . 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 11267 . . . . 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 11270 . . . 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 9072 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
20193ad2ant2 979 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  B  e.  CC )
213, 20addcomd 9260 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  +  B )  =  ( B  +  ( A  mod  C ) ) )
2221oveq1d 6088 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( A  mod  C )  +  B )  mod  C )  =  ( ( B  +  ( A  mod  C ) )  mod  C ) )
2318, 22eqtr4d 2470 . 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 11267 . . . 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 11270 . . 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 2471 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 1652    e. wcel 1725  (class class class)co 6073   CCcc 8980   RRcr 8981    + caddc 8985   RR+crp 10604    mod cmo 11242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fl 11194  df-mod 11243
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