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Theorem modsubdir 11008
Description: Distribute the modulo operation over a subtraction. (Contributed by NM, 30-Dec-2008.)
Assertion
Ref Expression
modsubdir  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( B  mod  C
)  <_  ( A  mod  C )  <->  ( ( A  -  B )  mod  C )  =  ( ( A  mod  C
)  -  ( B  mod  C ) ) ) )

Proof of Theorem modsubdir
StepHypRef Expression
1 modcl 10976 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( A  mod  C
)  e.  RR )
213adant2 974 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( A  mod  C )  e.  RR )
3 modcl 10976 . . . 4  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( B  mod  C
)  e.  RR )
433adant1 973 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( B  mod  C )  e.  RR )
52, 4subge0d 9362 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
0  <_  ( ( A  mod  C )  -  ( B  mod  C ) )  <->  ( B  mod  C )  <_  ( A  mod  C ) ) )
6 resubcl 9111 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
763adant3 975 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( A  -  B )  e.  RR )
8 simp3 957 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  C  e.  RR+ )
9 rerpdivcl 10381 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( A  /  C
)  e.  RR )
109flcld 10930 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( |_ `  ( A  /  C ) )  e.  ZZ )
11103adant2 974 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( |_ `  ( A  /  C ) )  e.  ZZ )
12 rerpdivcl 10381 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( B  /  C
)  e.  RR )
1312flcld 10930 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( |_ `  ( B  /  C ) )  e.  ZZ )
14133adant1 973 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( |_ `  ( B  /  C ) )  e.  ZZ )
1511, 14zsubcld 10122 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( |_ `  ( A  /  C ) )  -  ( |_ `  ( B  /  C
) ) )  e.  ZZ )
16 modcyc2 11000 . . . . . . 7  |-  ( ( ( A  -  B
)  e.  RR  /\  C  e.  RR+  /\  (
( |_ `  ( A  /  C ) )  -  ( |_ `  ( B  /  C
) ) )  e.  ZZ )  ->  (
( ( A  -  B )  -  ( C  x.  ( ( |_ `  ( A  /  C ) )  -  ( |_ `  ( B  /  C ) ) ) ) )  mod 
C )  =  ( ( A  -  B
)  mod  C )
)
177, 8, 15, 16syl3anc 1182 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( A  -  B )  -  ( C  x.  ( ( |_ `  ( A  /  C ) )  -  ( |_ `  ( B  /  C ) ) ) ) )  mod 
C )  =  ( ( A  -  B
)  mod  C )
)
18 recn 8827 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
19183ad2ant1 976 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  A  e.  CC )
20 recn 8827 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  e.  CC )
21203ad2ant2 977 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  B  e.  CC )
22 rpre 10360 . . . . . . . . . . . . 13  |-  ( C  e.  RR+  ->  C  e.  RR )
2322adantl 452 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  ->  C  e.  RR )
24 reflcl 10928 . . . . . . . . . . . . 13  |-  ( ( A  /  C )  e.  RR  ->  ( |_ `  ( A  /  C ) )  e.  RR )
259, 24syl 15 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( |_ `  ( A  /  C ) )  e.  RR )
2623, 25remulcld 8863 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( C  x.  ( |_ `  ( A  /  C ) ) )  e.  RR )
2726recnd 8861 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( C  x.  ( |_ `  ( A  /  C ) ) )  e.  CC )
28273adant2 974 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( C  x.  ( |_ `  ( A  /  C
) ) )  e.  CC )
2922adantl 452 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  ->  C  e.  RR )
30 reflcl 10928 . . . . . . . . . . . . 13  |-  ( ( B  /  C )  e.  RR  ->  ( |_ `  ( B  /  C ) )  e.  RR )
3112, 30syl 15 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( |_ `  ( B  /  C ) )  e.  RR )
3229, 31remulcld 8863 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( C  x.  ( |_ `  ( B  /  C ) ) )  e.  RR )
3332recnd 8861 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( C  x.  ( |_ `  ( B  /  C ) ) )  e.  CC )
34333adant1 973 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( C  x.  ( |_ `  ( B  /  C
) ) )  e.  CC )
3519, 21, 28, 34sub4d 9206 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  -  B
)  -  ( ( C  x.  ( |_
`  ( A  /  C ) ) )  -  ( C  x.  ( |_ `  ( B  /  C ) ) ) ) )  =  ( ( A  -  ( C  x.  ( |_ `  ( A  /  C ) ) ) )  -  ( B  -  ( C  x.  ( |_ `  ( B  /  C ) ) ) ) ) )
36223ad2ant3 978 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  C  e.  RR )
3736recnd 8861 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  C  e.  CC )
3825recnd 8861 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( |_ `  ( A  /  C ) )  e.  CC )
39383adant2 974 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( |_ `  ( A  /  C ) )  e.  CC )
4031recnd 8861 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( |_ `  ( B  /  C ) )  e.  CC )
41403adant1 973 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( |_ `  ( B  /  C ) )  e.  CC )
4237, 39, 41subdid 9235 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( C  x.  ( ( |_ `  ( A  /  C ) )  -  ( |_ `  ( B  /  C ) ) ) )  =  ( ( C  x.  ( |_ `  ( A  /  C ) ) )  -  ( C  x.  ( |_ `  ( B  /  C ) ) ) ) )
4342oveq2d 5874 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  -  B
)  -  ( C  x.  ( ( |_
`  ( A  /  C ) )  -  ( |_ `  ( B  /  C ) ) ) ) )  =  ( ( A  -  B )  -  (
( C  x.  ( |_ `  ( A  /  C ) ) )  -  ( C  x.  ( |_ `  ( B  /  C ) ) ) ) ) )
44 modval 10975 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( A  mod  C
)  =  ( A  -  ( C  x.  ( |_ `  ( A  /  C ) ) ) ) )
45443adant2 974 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( A  mod  C )  =  ( A  -  ( C  x.  ( |_ `  ( A  /  C
) ) ) ) )
46 modval 10975 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( B  mod  C
)  =  ( B  -  ( C  x.  ( |_ `  ( B  /  C ) ) ) ) )
47463adant1 973 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( B  mod  C )  =  ( B  -  ( C  x.  ( |_ `  ( B  /  C
) ) ) ) )
4845, 47oveq12d 5876 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  -  ( B  mod  C ) )  =  ( ( A  -  ( C  x.  ( |_ `  ( A  /  C ) ) ) )  -  ( B  -  ( C  x.  ( |_ `  ( B  /  C ) ) ) ) ) )
4935, 43, 483eqtr4d 2325 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  -  B
)  -  ( C  x.  ( ( |_
`  ( A  /  C ) )  -  ( |_ `  ( B  /  C ) ) ) ) )  =  ( ( A  mod  C )  -  ( B  mod  C ) ) )
5049oveq1d 5873 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( ( A  -  B )  -  ( C  x.  ( ( |_ `  ( A  /  C ) )  -  ( |_ `  ( B  /  C ) ) ) ) )  mod 
C )  =  ( ( ( A  mod  C )  -  ( B  mod  C ) )  mod  C ) )
5117, 50eqtr3d 2317 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  -  B
)  mod  C )  =  ( ( ( A  mod  C )  -  ( B  mod  C ) )  mod  C
) )
5251adantr 451 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  /\  0  <_  ( ( A  mod  C )  -  ( B  mod  C ) ) )  ->  (
( A  -  B
)  mod  C )  =  ( ( ( A  mod  C )  -  ( B  mod  C ) )  mod  C
) )
532, 4resubcld 9211 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  -  ( B  mod  C ) )  e.  RR )
5453adantr 451 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  /\  0  <_  ( ( A  mod  C )  -  ( B  mod  C ) ) )  ->  (
( A  mod  C
)  -  ( B  mod  C ) )  e.  RR )
55 simpl3 960 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  /\  0  <_  ( ( A  mod  C )  -  ( B  mod  C ) ) )  ->  C  e.  RR+ )
56 simpr 447 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  /\  0  <_  ( ( A  mod  C )  -  ( B  mod  C ) ) )  ->  0  <_  ( ( A  mod  C )  -  ( B  mod  C ) ) )
57 modge0 10980 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
0  <_  ( B  mod  C ) )
58573adant1 973 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  0  <_  ( B  mod  C
) )
592, 4subge02d 9364 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
0  <_  ( B  mod  C )  <->  ( ( A  mod  C )  -  ( B  mod  C ) )  <_  ( A  mod  C ) ) )
6058, 59mpbid 201 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  -  ( B  mod  C ) )  <_  ( A  mod  C ) )
61 modlt 10981 . . . . . . . 8  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( A  mod  C
)  <  C )
62613adant2 974 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( A  mod  C )  < 
C )
6353, 2, 36, 60, 62lelttrd 8974 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( A  mod  C
)  -  ( B  mod  C ) )  <  C )
6463adantr 451 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  /\  0  <_  ( ( A  mod  C )  -  ( B  mod  C ) ) )  ->  (
( A  mod  C
)  -  ( B  mod  C ) )  <  C )
65 modid 10993 . . . . 5  |-  ( ( ( ( ( A  mod  C )  -  ( B  mod  C ) )  e.  RR  /\  C  e.  RR+ )  /\  ( 0  <_  (
( A  mod  C
)  -  ( B  mod  C ) )  /\  ( ( A  mod  C )  -  ( B  mod  C ) )  <  C ) )  ->  ( (
( A  mod  C
)  -  ( B  mod  C ) )  mod  C )  =  ( ( A  mod  C )  -  ( B  mod  C ) ) )
6654, 55, 56, 64, 65syl22anc 1183 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  /\  0  <_  ( ( A  mod  C )  -  ( B  mod  C ) ) )  ->  (
( ( A  mod  C )  -  ( B  mod  C ) )  mod  C )  =  ( ( A  mod  C )  -  ( B  mod  C ) ) )
6752, 66eqtrd 2315 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  /\  0  <_  ( ( A  mod  C )  -  ( B  mod  C ) ) )  ->  (
( A  -  B
)  mod  C )  =  ( ( A  mod  C )  -  ( B  mod  C ) ) )
68 modge0 10980 . . . . . . 7  |-  ( ( ( A  -  B
)  e.  RR  /\  C  e.  RR+ )  -> 
0  <_  ( ( A  -  B )  mod  C ) )
696, 68sylan 457 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  RR+ )  ->  0  <_  (
( A  -  B
)  mod  C )
)
70693impa 1146 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  0  <_  ( ( A  -  B )  mod  C
) )
7170adantr 451 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  /\  ( ( A  -  B )  mod  C
)  =  ( ( A  mod  C )  -  ( B  mod  C ) ) )  -> 
0  <_  ( ( A  -  B )  mod  C ) )
72 simpr 447 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  /\  ( ( A  -  B )  mod  C
)  =  ( ( A  mod  C )  -  ( B  mod  C ) ) )  -> 
( ( A  -  B )  mod  C
)  =  ( ( A  mod  C )  -  ( B  mod  C ) ) )
7371, 72breqtrd 4047 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  /\  ( ( A  -  B )  mod  C
)  =  ( ( A  mod  C )  -  ( B  mod  C ) ) )  -> 
0  <_  ( ( A  mod  C )  -  ( B  mod  C ) ) )
7467, 73impbida 805 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
0  <_  ( ( A  mod  C )  -  ( B  mod  C ) )  <->  ( ( A  -  B )  mod 
C )  =  ( ( A  mod  C
)  -  ( B  mod  C ) ) ) )
755, 74bitr3d 246 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( B  mod  C
)  <_  ( A  mod  C )  <->  ( ( A  -  B )  mod  C )  =  ( ( A  mod  C
)  -  ( B  mod  C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   ZZcz 10024   RR+crp 10354   |_cfl 10924    mod cmo 10973
This theorem is referenced by:  digit1  11235  4sqlem12  13003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974
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