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Theorem q2submod 10189
Description: If a number is between a modulus and twice the modulus, the first number modulo the modulus equals the first number minus the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
Assertion
Ref Expression
q2submod  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  mod  B )  =  ( A  -  B ) )

Proof of Theorem q2submod
StepHypRef Expression
1 qcn 9453 . . . . . . 7  |-  ( B  e.  QQ  ->  B  e.  CC )
213ad2ant2 1004 . . . . . 6  |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  ->  B  e.  CC )
32adantr 274 . . . . 5  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  B  e.  CC )
43mulid1d 7807 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( B  x.  1 )  =  B )
54oveq2d 5798 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  -  ( B  x.  1
) )  =  ( A  -  B ) )
65oveq1d 5797 . 2  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( ( A  -  ( B  x.  1 ) )  mod 
B )  =  ( ( A  -  B
)  mod  B )
)
7 simpl1 985 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  A  e.  QQ )
8 1zzd 9105 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  1  e.  ZZ )
9 simpl2 986 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  B  e.  QQ )
10 simpl3 987 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  0  <  B
)
11 modqcyc2 10164 . . 3  |-  ( ( ( A  e.  QQ  /\  1  e.  ZZ )  /\  ( B  e.  QQ  /\  0  < 
B ) )  -> 
( ( A  -  ( B  x.  1
) )  mod  B
)  =  ( A  mod  B ) )
127, 8, 9, 10, 11syl22anc 1218 . 2  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( ( A  -  ( B  x.  1 ) )  mod 
B )  =  ( A  mod  B ) )
13 qsubcl 9457 . . . 4  |-  ( ( A  e.  QQ  /\  B  e.  QQ )  ->  ( A  -  B
)  e.  QQ )
147, 9, 13syl2anc 409 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  -  B )  e.  QQ )
15 simpr 109 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( B  <_  A  /\  A  <  (
2  x.  B ) ) )
16 qre 9444 . . . . . . . 8  |-  ( A  e.  QQ  ->  A  e.  RR )
177, 16syl 14 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  A  e.  RR )
18 qre 9444 . . . . . . . 8  |-  ( B  e.  QQ  ->  B  e.  RR )
199, 18syl 14 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  B  e.  RR )
2017, 19subge0d 8321 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( 0  <_ 
( A  -  B
)  <->  B  <_  A ) )
2120bicomd 140 . . . . 5  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( B  <_  A 
<->  0  <_  ( A  -  B ) ) )
2232timesd 8986 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( 2  x.  B )  =  ( B  +  B ) )
2322breq2d 3949 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  < 
( 2  x.  B
)  <->  A  <  ( B  +  B ) ) )
2417, 19, 19ltsubaddd 8327 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( ( A  -  B )  < 
B  <->  A  <  ( B  +  B ) ) )
2523, 24bitr4d 190 . . . . 5  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  < 
( 2  x.  B
)  <->  ( A  -  B )  <  B
) )
2621, 25anbi12d 465 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( ( B  <_  A  /\  A  <  ( 2  x.  B
) )  <->  ( 0  <_  ( A  -  B )  /\  ( A  -  B )  <  B ) ) )
2715, 26mpbid 146 . . 3  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( 0  <_ 
( A  -  B
)  /\  ( A  -  B )  <  B
) )
28 modqid 10153 . . 3  |-  ( ( ( ( A  -  B )  e.  QQ  /\  B  e.  QQ )  /\  ( 0  <_ 
( A  -  B
)  /\  ( A  -  B )  <  B
) )  ->  (
( A  -  B
)  mod  B )  =  ( A  -  B ) )
2914, 9, 27, 28syl21anc 1216 . 2  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( ( A  -  B )  mod 
B )  =  ( A  -  B ) )
306, 12, 293eqtr3d 2181 1  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  mod  B )  =  ( A  -  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    = wceq 1332    e. wcel 1481   class class class wbr 3937  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645    + caddc 7647    x. cmul 7649    < clt 7824    <_ cle 7825    - cmin 7957   2c2 8795   ZZcz 9078   QQcq 9438    mod cmo 10126
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-n0 9002  df-z 9079  df-q 9439  df-rp 9471  df-fl 10074  df-mod 10127
This theorem is referenced by:  modifeq2int  10190  modaddmodup  10191
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