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Theorem modsubdir 11024
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 10992 . . . 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 10992 . . . 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 9378 . 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 9127 . . . . . . . 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 10397 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( A  /  C
)  e.  RR )
109flcld 10946 . . . . . . . . 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 10397 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( B  /  C
)  e.  RR )
1312flcld 10946 . . . . . . . . 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 10138 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  (
( |_ `  ( A  /  C ) )  -  ( |_ `  ( B  /  C
) ) )  e.  ZZ )
16 modcyc2 11016 . . . . . . 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 8843 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
19183ad2ant1 976 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  A  e.  CC )
20 recn 8843 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  e.  CC )
21203ad2ant2 977 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  B  e.  CC )
22 rpre 10376 . . . . . . . . . . . . 13  |-  ( C  e.  RR+  ->  C  e.  RR )
2322adantl 452 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  ->  C  e.  RR )
24 reflcl 10944 . . . . . . . . . . . . 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 8879 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  C  e.  RR+ )  -> 
( C  x.  ( |_ `  ( A  /  C ) ) )  e.  RR )
2726recnd 8877 . . . . . . . . . 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 10944 . . . . . . . . . . . . 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 8879 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  C  e.  RR+ )  -> 
( C  x.  ( |_ `  ( B  /  C ) ) )  e.  RR )
3332recnd 8877 . . . . . . . . . 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 9222 . . . . . . . 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 8877 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  C  e.  CC )
3825recnd 8877 . . . . . . . . . . 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 8877 . . . . . . . . . . 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 9251 . . . . . . . . 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 5890 . . . . . . . 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 10991 . . . . . . . . . 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 10991 . . . . . . . . . 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 5892 . . . . . . . 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 2338 . . . . . . 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 5889 . . . . . 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 2330 . . . . 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 9227 . . . . . 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 10996 . . . . . . . . 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 9380 . . . . . . . 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 10997 . . . . . . . 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 8990 . . . . . 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 11009 . . . . 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 2328 . . 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 10996 . . . . . . 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 4063 . . 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   ZZcz 10040   RR+crp 10370   |_cfl 10940    mod cmo 10989
This theorem is referenced by:  digit1  11251  4sqlem12  13019
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-mod 10990
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