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Theorem modqaddabs 9830
Description: Absorption law for modulo. (Contributed by Jim Kingdon, 22-Oct-2021.)
Assertion
Ref Expression
modqaddabs |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( ( A mod C ) + ( B mod C ) ) mod C ) = ( ( A + B ) mod C ) )
Proof of Theorem modqaddabs
StepHypRef Expression
1 simpll 497 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> A e. QQ )
2 simprl 499 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> C e. QQ )
3 simprr 500 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> 0 < C )
41, 2, 3modqcld 9796 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( A mod C ) e. QQ )
5 qcn 9180 . . . . 5 |- ( ( A mod C ) e. QQ -> ( A mod C ) e. CC )
64, 5syl 14 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( A mod C ) e. CC )
7 simplr 498 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> B e. QQ )
87, 2, 3modqcld 9796 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( B mod C ) e. QQ )
9 qcn 9180 . . . . 5 |- ( ( B mod C ) e. QQ -> ( B mod C ) e. CC )
108, 9syl 14 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( B mod C ) e. CC )
116, 10addcomd 7694 . . 3 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( A mod C ) + ( B mod C ) ) = ( ( B mod C ) + ( A mod C ) ) )
1211oveq1d 5681 . 2 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( ( A mod C ) + ( B mod C ) ) mod C ) = ( ( ( B mod C ) + ( A mod C ) ) mod C ) )
13 modqabs2 9826 . . . . 5 |- ( ( B e. QQ /\ C e. QQ /\ 0 < C ) -> ( ( B mod C ) mod C ) = ( B mod C ) )
147, 2, 3, 13syl3anc 1175 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( B mod C ) mod C ) = ( B mod C ) )
158, 7, 4, 2, 3, 14modqadd1 9829 . . 3 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( ( B mod C ) + ( A mod C ) ) mod C ) = ( ( B + ( A mod C ) ) mod C ) )
16 qcn 9180 . . . . . 6 |- ( B e. QQ -> B e. CC )
177, 16syl 14 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> B e. CC )
186, 17addcomd 7694 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( A mod C ) + B ) = ( B + ( A mod C ) ) )
1918oveq1d 5681 . . 3 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( ( A mod C ) + B ) mod C ) = ( ( B + ( A mod C ) ) mod C ) )
2015, 19eqtr4d 2124 . 2 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( ( B mod C ) + ( A mod C ) ) mod C ) = ( ( ( A mod C ) + B ) mod C ) )
21 modqabs2 9826 . . . 4 |- ( ( A e. QQ /\ C e. QQ /\ 0 < C ) -> ( ( A mod C ) mod C ) = ( A mod C ) )
221, 2, 3, 21syl3anc 1175 . . 3 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( A mod C ) mod C ) = ( A mod C ) )
234, 1, 7, 2, 3, 22modqadd1 9829 . 2 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( ( A mod C ) + B ) mod C ) = ( ( A + B ) mod C ) )
2412, 20, 233eqtrd 2125 1 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( ( A mod C ) + ( B mod C ) ) mod C ) = ( ( A + B ) mod C ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   class class class wbr 3851  (class class class)co 5666   CCcc 7409   0cc0 7411   + caddc 7414   < clt 7583   QQcq 9165   mod cmo 9790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-po 4132  df-iso 4133  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-n0 8735  df-z 8812  df-q 9166  df-rp 9196  df-fl 9738  df-mod 9791
This theorem is referenced by:  modfsummodlemstep  10912
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