ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  modqaddmulmod Unicode version
Theorem modqaddmulmod 10654
Description: The sum of a rational number and the product of a second rational number modulo a modulus and an integer equals the sum of the rational number and the product of the other rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
Assertion
Ref Expression
modqaddmulmod |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + ( ( B mod M ) x. C ) ) mod M ) = ( ( A + ( B x. C ) ) mod M ) )
Proof of Theorem modqaddmulmod
StepHypRef Expression
1 simpl1 1026 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. QQ )
2 qcn 9868 . . . . 5 |- ( A e. QQ -> A e. CC )
31, 2syl 14 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. CC )
4 simpl2 1027 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> B e. QQ )
5 simprl 531 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> M e. QQ )
6 simprr 533 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> 0 < M )
74, 5, 6modqcld 10591 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B mod M ) e. QQ )
8 simpl3 1028 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> C e. ZZ )
9 zq 9860 . . . . . . 7 |- ( C e. ZZ -> C e. QQ )
108, 9syl 14 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> C e. QQ )
11 qmulcl 9871 . . . . . 6 |- ( ( ( B mod M ) e. QQ /\ C e. QQ ) -> ( ( B mod M ) x. C ) e. QQ )
127, 10, 11syl2anc 411 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B mod M ) x. C ) e. QQ )
13 qcn 9868 . . . . 5 |- ( ( ( B mod M ) x. C ) e. QQ -> ( ( B mod M ) x. C ) e. CC )
1412, 13syl 14 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B mod M ) x. C ) e. CC )
153, 14addcomd 8330 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( A + ( ( B mod M ) x. C ) ) = ( ( ( B mod M ) x. C ) + A ) )
1615oveq1d 6033 . 2 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + ( ( B mod M ) x. C ) ) mod M ) = ( ( ( ( B mod M ) x. C ) + A ) mod M ) )
1793ad2ant3 1046 . . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) -> C e. QQ )
1817adantr 276 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> C e. QQ )
197, 18, 11syl2anc 411 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B mod M ) x. C ) e. QQ )
20 qmulcl 9871 . . . . 5 |- ( ( B e. QQ /\ C e. QQ ) -> ( B x. C ) e. QQ )
214, 18, 20syl2anc 411 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B x. C ) e. QQ )
2221, 5, 6modqcld 10591 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B x. C ) mod M ) e. QQ )
23 modqmulmod 10652 . . . . 5 |- ( ( ( B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B mod M ) x. C ) mod M ) = ( ( B x. C ) mod M ) )
24233adantl1 1179 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B mod M ) x. C ) mod M ) = ( ( B x. C ) mod M ) )
25 modqabs2 10621 . . . . 5 |- ( ( ( B x. C ) e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( ( B x. C ) mod M ) mod M ) = ( ( B x. C ) mod M ) )
2621, 5, 6, 25syl3anc 1273 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B x. C ) mod M ) mod M ) = ( ( B x. C ) mod M ) )
2724, 26eqtr4d 2267 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B mod M ) x. C ) mod M ) = ( ( ( B x. C ) mod M ) mod M ) )
2819, 22, 1, 5, 6, 27modqadd1 10624 . 2 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( ( B mod M ) x. C ) + A ) mod M ) = ( ( ( ( B x. C ) mod M ) + A ) mod M ) )
29 modqaddmod 10626 . . . 4 |- ( ( ( ( B x. C ) e. QQ /\ A e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( ( B x. C ) mod M ) + A ) mod M ) = ( ( ( B x. C ) + A ) mod M ) )
3021, 1, 5, 6, 29syl22anc 1274 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( ( B x. C ) mod M ) + A ) mod M ) = ( ( ( B x. C ) + A ) mod M ) )
31 qcn 9868 . . . . . 6 |- ( ( B x. C ) e. QQ -> ( B x. C ) e. CC )
3221, 31syl 14 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( B x. C ) e. CC )
3332, 3addcomd 8330 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( B x. C ) + A ) = ( A + ( B x. C ) ) )
3433oveq1d 6033 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( B x. C ) + A ) mod M ) = ( ( A + ( B x. C ) ) mod M ) )
3530, 34eqtrd 2264 . 2 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( ( B x. C ) mod M ) + A ) mod M ) = ( ( A + ( B x. C ) ) mod M ) )
3616, 28, 353eqtrd 2268 1 |- ( ( ( A e. QQ /\ B e. QQ /\ C e. ZZ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + ( ( B mod M ) x. C ) ) mod M ) = ( ( A + ( B x. C ) ) mod M ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6018   CCcc 8030   0cc0 8032   + caddc 8035   x. cmul 8037   < clt 8214   ZZcz 9479   QQcq 9853   mod cmo 10585
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151
This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-n0 9403  df-z 9480  df-q 9854  df-rp 9889  df-fl 10531  df-mod 10586
This theorem is referenced by:  modprm0  12832  modprmn0modprm0  12834
  Copyright terms: Public domain W3C validator