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Theorem q2submod 10747
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 9966 . . . . . . 7 |- ( B e. QQ -> B e. CC )
213ad2ant2 1046 . . . . . 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 )
43mulridd 8291 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( B x. 1 ) = B )
54oveq2d 6066 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A - ( B x. 1 ) ) = ( A - B ) )
65oveq1d 6065 . 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 1027 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> A e. QQ )
8 1zzd 9604 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> 1 e. ZZ )
9 simpl2 1028 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> B e. QQ )
10 simpl3 1029 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> 0 < B )
11 modqcyc2 10722 . . 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 1275 . 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 9970 . . . 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 9957 . . . . . . . 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 9957 . . . . . . . 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 8809 . . . . . 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 9481 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( 2 x. B ) = ( B + B ) )
2322breq2d 4121 . . . . . 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 8815 . . . . . 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 10711 . . 3 |- ( ( ( ( A - B ) e. QQ /\ B e. QQ ) /\ ( 0 <_ ( A - B ) /\ ( A - B ) < B ) ) -> ( ( A - B ) mod B ) = ( A - B ) )
2914, 9, 27, 28syl21anc 1273 . 2 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( ( A - B ) mod B ) = ( A - B ) )
306, 12, 293eqtr3d 2273 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 1005   = wceq 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   < clt 8308   <_ cle 8309   - cmin 8444   2c2 9288   ZZcz 9577   QQcq 9951   mod cmo 10684
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246
This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-q 9952  df-rp 9987  df-fl 10630  df-mod 10685
This theorem is referenced by:  modifeq2int  10748  modaddmodup  10749
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