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Theorem modqaddabs 10520
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 10486 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( A mod C ) e. QQ )
5 qcn 9768 . . . . 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 10486 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( B mod C ) e. QQ )
9 qcn 9768 . . . . 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 8236 . . 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 5969 . 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 10516 . . . . 5 |- ( ( B e. QQ /\ C e. QQ /\ 0 < C ) -> ( ( B mod C ) mod C ) = ( B mod C ) )
147, 2, 3, 13syl3anc 1250 . . . 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 10519 . . 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 9768 . . . . . 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 8236 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( C e. QQ /\ 0 < C ) ) -> ( ( A mod C ) + B ) = ( B + ( A mod C ) ) )
1918oveq1d 5969 . . 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 2242 . 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 10516 . . . 4 |- ( ( A e. QQ /\ C e. QQ /\ 0 < C ) -> ( ( A mod C ) mod C ) = ( A mod C ) )
221, 2, 3, 21syl3anc 1250 . . 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 10519 . 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 2243 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 1373   e. wcel 2177   class class class wbr 4048  (class class class)co 5954   CCcc 7936   0cc0 7938   + caddc 7941   < clt 8120   QQcq 9753   mod cmo 10480
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-mulrcl 8037  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-0lt1 8044  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-precex 8048  ax-cnre 8049  ax-pre-ltirr 8050  ax-pre-ltwlin 8051  ax-pre-lttrn 8052  ax-pre-apti 8053  ax-pre-ltadd 8054  ax-pre-mulgt0 8055  ax-pre-mulext 8056  ax-arch 8057
This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-po 4348  df-iso 4349  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-pnf 8122  df-mnf 8123  df-xr 8124  df-ltxr 8125  df-le 8126  df-sub 8258  df-neg 8259  df-reap 8661  df-ap 8668  df-div 8759  df-inn 9050  df-n0 9309  df-z 9386  df-q 9754  df-rp 9789  df-fl 10426  df-mod 10481
This theorem is referenced by:  modfsummodlemstep  11818
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