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Theorem modqdi 10365
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 1021 . . . . 5  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  A  e.  QQ )
2 qcn 9610 . . . . 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 998 . . . . 5  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  B  e.  QQ )
5 qcn 9610 . . . . 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 1025 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  C  e.  QQ )
8 simp3r 1026 . . . . . . . . . 10  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  0  <  C )
98gt0ne0d 8446 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  C  =/=  0 )
10 qdivcl 9619 . . . . . . . . 9  |-  ( ( B  e.  QQ  /\  C  e.  QQ  /\  C  =/=  0 )  ->  ( B  /  C )  e.  QQ )
114, 7, 9, 10syl3anc 1238 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( B  /  C )  e.  QQ )
1211flqcld 10250 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( |_ `  ( B  /  C
) )  e.  ZZ )
13 zq 9602 . . . . . . 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 9613 . . . . . 6  |-  ( ( C  e.  QQ  /\  ( |_ `  ( B  /  C ) )  e.  QQ )  -> 
( C  x.  ( |_ `  ( B  /  C ) ) )  e.  QQ )
167, 14, 15syl2anc 411 . . . . 5  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( C  x.  ( |_ `  ( B  /  C ) ) )  e.  QQ )
17 qcn 9610 . . . . 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 8348 . . 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 9610 . . . . . . . . 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 9601 . . . . . . . . . 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 8563 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  C #  0
)
25 qre 9601 . . . . . . . . . 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 1022 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  0  <  A )
2826, 27gt0ap0d 8563 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  A #  0
)
296, 21, 3, 24, 28divcanap5d 8750 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( ( A  x.  B )  /  ( A  x.  C ) )  =  ( B  /  C
) )
3029fveq2d 5514 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( |_ `  ( ( A  x.  B )  /  ( A  x.  C )
) )  =  ( |_ `  ( B  /  C ) ) )
3130oveq2d 5884 . . . . 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 9352 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( |_ `  ( B  /  C
) )  e.  CC )
333, 21, 32mulassd 7958 . . . . 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 2211 . . . 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 5884 . . 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 2210 . 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 10297 . . . 4  |-  ( ( B  e.  QQ  /\  C  e.  QQ  /\  0  <  C )  ->  ( B  mod  C )  =  ( B  -  ( C  x.  ( |_ `  ( B  /  C
) ) ) ) )
384, 7, 8, 37syl3anc 1238 . . 3  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( B  mod  C )  =  ( B  -  ( C  x.  ( |_ `  ( B  /  C
) ) ) ) )
3938oveq2d 5884 . 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 9613 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  x.  B
)  e.  QQ )
411, 4, 40syl2anc 411 . . 3  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  B )  e.  QQ )
42 qmulcl 9613 . . . 4  |-  ( ( A  e.  QQ  /\  C  e.  QQ )  ->  ( A  x.  C
)  e.  QQ )
431, 7, 42syl2anc 411 . . 3  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  C )  e.  QQ )
4426, 23, 27, 8mulgt0d 8057 . . 3  |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  0  <  ( A  x.  C ) )
45 modqval 10297 . . 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 1238 . 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 2220 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 104    /\ w3a 978    = wceq 1353    e. wcel 2148    =/= wne 2347   class class class wbr 4000   ` cfv 5211  (class class class)co 5868   CCcc 7787   RRcr 7788   0cc0 7789    x. cmul 7794    < clt 7969    - cmin 8105    / cdiv 8605   ZZcz 9229   QQcq 9595   |_cfl 10241    mod cmo 10295
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907  ax-arch 7908
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-po 4292  df-iso 4293  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-n0 9153  df-z 9230  df-q 9596  df-rp 9628  df-fl 10243  df-mod 10296
This theorem is referenced by: (None)
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