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Theorem q2submod 10371
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 9623 . . . . . . 7 |- ( B e. QQ -> B e. CC )
213ad2ant2 1019 . . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. CC )
32adantr 276 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> B e. CC )
43mulid1d 7965 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( B x. 1 ) = B )
54oveq2d 5885 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A - ( B x. 1 ) ) = ( A - B ) )
65oveq1d 5884 . 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 1000 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> A e. QQ )
8 1zzd 9269 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> 1 e. ZZ )
9 simpl2 1001 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> B e. QQ )
10 simpl3 1002 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> 0 < B )
11 modqcyc2 10346 . . 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 1239 . 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 9627 . . . 4 |- ( ( A e. QQ /\ B e. QQ ) -> ( A - B ) e. QQ )
147, 9, 13syl2anc 411 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A - B ) e. QQ )
15 simpr 110 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( B <_ A /\ A < ( 2 x. B ) ) )
16 qre 9614 . . . . . . . 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 9614 . . . . . . . 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 8482 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( 0 <_ ( A - B ) <-> B <_ A ) )
2120bicomd 141 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( B <_ A <-> 0 <_ ( A - B ) ) )
2232timesd 9150 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( 2 x. B ) = ( B + B ) )
2322breq2d 4012 . . . . . 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 8488 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( ( A - B ) < B <-> A < ( B + B ) ) )
2523, 24bitr4d 191 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A < ( 2 x. B ) <-> ( A - B ) < B ) )
2621, 25anbi12d 473 . . . 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 147 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( 0 <_ ( A - B ) /\ ( A - B ) < B ) )
28 modqid 10335 . . 3 |- ( ( ( ( A - B ) e. QQ /\ B e. QQ ) /\ ( 0 <_ ( A - B ) /\ ( A - B ) < B ) ) -> ( ( A - B ) mod B ) = ( A - B ) )
2914, 9, 27, 28syl21anc 1237 . 2 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( ( A - B ) mod B ) = ( A - B ) )
306, 12, 293eqtr3d 2218 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 104   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4000  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   < clt 7982   <_ cle 7983   - cmin 8118   2c2 8959   ZZcz 9242   QQcq 9608   mod cmo 10308
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 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921
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 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-q 9609  df-rp 9641  df-fl 10256  df-mod 10309
This theorem is referenced by:  modifeq2int  10372  modaddmodup  10373
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