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Theorem modqdi 10133
Description: Distribute multiplication over a modulo operation. (Contributed by Jim Kingdon, 26-Oct-2021.)
Assertion
Ref Expression
modqdi  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  ( B  mod  C
) )  =  ( ( A  x.  B
)  mod  ( A  x.  C ) ) )

Proof of Theorem modqdi
StepHypRef Expression
1 simp1l 990 . . . . 5  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  A  e.  QQ )
2 qcn 9394 . . . . 5  |-  ( A  e.  QQ  ->  A  e.  CC )
31, 2syl 14 . . . 4  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  A  e.  CC )
4 simp2 967 . . . . 5  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  B  e.  QQ )
5 qcn 9394 . . . . 5  |-  ( B  e.  QQ  ->  B  e.  CC )
64, 5syl 14 . . . 4  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  B  e.  CC )
7 simp3l 994 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  C  e.  QQ )
8 simp3r 995 . . . . . . . . . 10  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  0  <  C )
98gt0ne0d 8242 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  C  =/=  0 )
10 qdivcl 9403 . . . . . . . . 9  |-  ( ( B  e.  QQ  /\  C  e.  QQ  /\  C  =/=  0 )  ->  ( B  /  C )  e.  QQ )
114, 7, 9, 10syl3anc 1201 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( B  /  C )  e.  QQ )
1211flqcld 10018 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( |_ `  ( B  /  C
) )  e.  ZZ )
13 zq 9386 . . . . . . 7  |-  ( ( |_ `  ( B  /  C ) )  e.  ZZ  ->  ( |_ `  ( B  /  C ) )  e.  QQ )
1412, 13syl 14 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( |_ `  ( B  /  C
) )  e.  QQ )
15 qmulcl 9397 . . . . . 6  |-  ( ( C  e.  QQ  /\  ( |_ `  ( B  /  C ) )  e.  QQ )  -> 
( C  x.  ( |_ `  ( B  /  C ) ) )  e.  QQ )
167, 14, 15syl2anc 408 . . . . 5  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( C  x.  ( |_ `  ( B  /  C ) ) )  e.  QQ )
17 qcn 9394 . . . . 5  |-  ( ( C  x.  ( |_
`  ( B  /  C ) ) )  e.  QQ  ->  ( C  x.  ( |_ `  ( B  /  C
) ) )  e.  CC )
1816, 17syl 14 . . . 4  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( C  x.  ( |_ `  ( B  /  C ) ) )  e.  CC )
193, 6, 18subdid 8144 . . 3  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  ( B  -  ( C  x.  ( |_ `  ( B  /  C
) ) ) ) )  =  ( ( A  x.  B )  -  ( A  x.  ( C  x.  ( |_ `  ( B  /  C ) ) ) ) ) )
20 qcn 9394 . . . . . . . . 9  |-  ( C  e.  QQ  ->  C  e.  CC )
217, 20syl 14 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  C  e.  CC )
22 qre 9385 . . . . . . . . . 10  |-  ( C  e.  QQ  ->  C  e.  RR )
237, 22syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  C  e.  RR )
2423, 8gt0ap0d 8359 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  C #  0
)
25 qre 9385 . . . . . . . . . 10  |-  ( A  e.  QQ  ->  A  e.  RR )
261, 25syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  A  e.  RR )
27 simp1r 991 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  0  <  A )
2826, 27gt0ap0d 8359 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  A #  0
)
296, 21, 3, 24, 28divcanap5d 8545 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( ( A  x.  B )  /  ( A  x.  C ) )  =  ( B  /  C
) )
3029fveq2d 5393 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( |_ `  ( ( A  x.  B )  /  ( A  x.  C )
) )  =  ( |_ `  ( B  /  C ) ) )
3130oveq2d 5758 . . . . 5  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( ( A  x.  C )  x.  ( |_ `  (
( A  x.  B
)  /  ( A  x.  C ) ) ) )  =  ( ( A  x.  C
)  x.  ( |_
`  ( B  /  C ) ) ) )
3212zcnd 9142 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( |_ `  ( B  /  C
) )  e.  CC )
333, 21, 32mulassd 7757 . . . . 5  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( ( A  x.  C )  x.  ( |_ `  ( B  /  C ) ) )  =  ( A  x.  ( C  x.  ( |_ `  ( B  /  C ) ) ) ) )
3431, 33eqtr2d 2151 . . . 4  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  ( C  x.  ( |_ `  ( B  /  C ) ) ) )  =  ( ( A  x.  C )  x.  ( |_ `  ( ( A  x.  B )  /  ( A  x.  C )
) ) ) )
3534oveq2d 5758 . . 3  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( ( A  x.  B )  -  ( A  x.  ( C  x.  ( |_ `  ( B  /  C ) ) ) ) )  =  ( ( A  x.  B
)  -  ( ( A  x.  C )  x.  ( |_ `  ( ( A  x.  B )  /  ( A  x.  C )
) ) ) ) )
3619, 35eqtrd 2150 . 2  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  ( B  -  ( C  x.  ( |_ `  ( B  /  C
) ) ) ) )  =  ( ( A  x.  B )  -  ( ( A  x.  C )  x.  ( |_ `  (
( A  x.  B
)  /  ( A  x.  C ) ) ) ) ) )
37 modqval 10065 . . . 4  |-  ( ( B  e.  QQ  /\  C  e.  QQ  /\  0  <  C )  ->  ( B  mod  C )  =  ( B  -  ( C  x.  ( |_ `  ( B  /  C
) ) ) ) )
384, 7, 8, 37syl3anc 1201 . . 3  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( B  mod  C )  =  ( B  -  ( C  x.  ( |_ `  ( B  /  C
) ) ) ) )
3938oveq2d 5758 . 2  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  ( B  mod  C
) )  =  ( A  x.  ( B  -  ( C  x.  ( |_ `  ( B  /  C ) ) ) ) ) )
40 qmulcl 9397 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  x.  B
)  e.  QQ )
411, 4, 40syl2anc 408 . . 3  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  B )  e.  QQ )
42 qmulcl 9397 . . . 4  |-  ( ( A  e.  QQ  /\  C  e.  QQ )  ->  ( A  x.  C
)  e.  QQ )
431, 7, 42syl2anc 408 . . 3  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  C )  e.  QQ )
4426, 23, 27, 8mulgt0d 7853 . . 3  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  0  <  ( A  x.  C ) )
45 modqval 10065 . . 3  |-  ( ( ( A  x.  B
)  e.  QQ  /\  ( A  x.  C
)  e.  QQ  /\  0  <  ( A  x.  C ) )  -> 
( ( A  x.  B )  mod  ( A  x.  C )
)  =  ( ( A  x.  B )  -  ( ( A  x.  C )  x.  ( |_ `  (
( A  x.  B
)  /  ( A  x.  C ) ) ) ) ) )
4641, 43, 44, 45syl3anc 1201 . 2  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( ( A  x.  B )  mod  ( A  x.  C
) )  =  ( ( A  x.  B
)  -  ( ( A  x.  C )  x.  ( |_ `  ( ( A  x.  B )  /  ( A  x.  C )
) ) ) ) )
4736, 39, 463eqtr4d 2160 1  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  ( B  mod  C
) )  =  ( ( A  x.  B
)  mod  ( A  x.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316    e. wcel 1465    =/= wne 2285   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   CCcc 7586   RRcr 7587   0cc0 7588    x. cmul 7593    < clt 7768    - cmin 7901    / cdiv 8400   ZZcz 9022   QQcq 9379   |_cfl 10009    mod cmo 10063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-n0 8946  df-z 9023  df-q 9380  df-rp 9410  df-fl 10011  df-mod 10064
This theorem is referenced by: (None)
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