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Theorem modqaddabs 10571
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 527 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> A e. QQ )
2 simprl 529 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> C e. QQ )
3 simprr 531 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> 0 < C )
41, 2, 3modqcld 10537 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( A mod C ) e. QQ )
5 qcn 9817 . . . . 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 528 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> B e. QQ )
87, 2, 3modqcld 10537 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( B mod C ) e. QQ )
9 qcn 9817 . . . . 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 8285 . . 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 6009 . 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 10567 . . . . 5 |- ( ( B e. QQ /\ C e. QQ /\ 0 < C ) -> ( ( B mod C ) mod C ) = ( B mod C ) )
147, 2, 3, 13syl3anc 1271 . . . 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 10570 . . 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 9817 . . . . . 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 8285 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( A mod C ) + B ) = ( B + ( A mod C ) ) )
1918oveq1d 6009 . . 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 2265 . 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 10567 . . . 4 |- ( ( A e. QQ /\ C e. QQ /\ 0 < C ) -> ( ( A mod C ) mod C ) = ( A mod C ) )
221, 2, 3, 21syl3anc 1271 . . 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 10570 . 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 2266 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 104   = wceq 1395   e. wcel 2200   class class class wbr 4082  (class class class)co 5994   CCcc 7985   0cc0 7987   + caddc 7990   < clt 8169   QQcq 9802   mod cmo 10531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106
This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-po 4384  df-iso 4385  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-n0 9358  df-z 9435  df-q 9803  df-rp 9838  df-fl 10477  df-mod 10532
This theorem is referenced by:  modfsummodlemstep  11954
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