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Theorem q2submod 10320
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 9572 . . . . . . 7 |- ( B e. QQ -> B e. CC )
213ad2ant2 1009 . . . . . 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 7916 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( B x. 1 ) = B )
54oveq2d 5858 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A - ( B x. 1 ) ) = ( A - B ) )
65oveq1d 5857 . 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 990 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> A e. QQ )
8 1zzd 9218 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> 1 e. ZZ )
9 simpl2 991 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> B e. QQ )
10 simpl3 992 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> 0 < B )
11 modqcyc2 10295 . . 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 1229 . 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 9576 . . . 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 9563 . . . . . . . 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 9563 . . . . . . . 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 8433 . . . . . 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 9099 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( 2 x. B ) = ( B + B ) )
2322breq2d 3994 . . . . . 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 8439 . . . . . 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 10284 . . 3 |- ( ( ( ( A - B ) e. QQ /\ B e. QQ ) /\ ( 0 <_ ( A - B ) /\ ( A - B ) < B ) ) -> ( ( A - B ) mod B ) = ( A - B ) )
2914, 9, 27, 28syl21anc 1227 . 2 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( ( A - B ) mod B ) = ( A - B ) )
306, 12, 293eqtr3d 2206 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 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   < clt 7933   <_ cle 7934   - cmin 8069   2c2 8908   ZZcz 9191   QQcq 9557   mod cmo 10257
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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872
This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192  df-q 9558  df-rp 9590  df-fl 10205  df-mod 10258
This theorem is referenced by:  modifeq2int  10321  modaddmodup  10322
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