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Theorem modqaddabs 10392
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 10358 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( A mod C ) e. QQ )
5 qcn 9663 . . . . 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 10358 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( B mod C ) e. QQ )
9 qcn 9663 . . . . 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 8137 . . 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 5910 . 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 10388 . . . . 5 |- ( ( B e. QQ /\ C e. QQ /\ 0 < C ) -> ( ( B mod C ) mod C ) = ( B mod C ) )
147, 2, 3, 13syl3anc 1249 . . . 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 10391 . . 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 9663 . . . . . 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 8137 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( A mod C ) + B ) = ( B + ( A mod C ) ) )
1918oveq1d 5910 . . 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 2225 . 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 10388 . . . 4 |- ( ( A e. QQ /\ C e. QQ /\ 0 < C ) -> ( ( A mod C ) mod C ) = ( A mod C ) )
221, 2, 3, 21syl3anc 1249 . . 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 10391 . 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 2226 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 1364   e. wcel 2160   class class class wbr 4018  (class class class)co 5895   CCcc 7838   0cc0 7840   + caddc 7843   < clt 8021   QQcq 9648   mod cmo 10352
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-n0 9206  df-z 9283  df-q 9649  df-rp 9683  df-fl 10300  df-mod 10353
This theorem is referenced by:  modfsummodlemstep  11496
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