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Theorem q2submod 10800
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 10013 . . . . . . 7 |- ( B e. QQ -> B e. CC )
213ad2ant2 1050 . . . . . 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 8333 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( B x. 1 ) = B )
54oveq2d 6091 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A - ( B x. 1 ) ) = ( A - B ) )
65oveq1d 6090 . 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 1031 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> A e. QQ )
8 1zzd 9650 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> 1 e. ZZ )
9 simpl2 1032 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> B e. QQ )
10 simpl3 1033 . . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> 0 < B )
11 modqcyc2 10775 . . 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 1279 . 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 10017 . . . 4 |- ( ( A e. QQ /\ B e. QQ ) -> ( A - B ) e. QQ )
147, 9, 13syl2anc 415 . . 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 10004 . . . . . . . 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 10004 . . . . . . . 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 8853 . . . . . 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 9527 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( 2 x. B ) = ( B + B ) )
2322breq2d 4137 . . . . . 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 8859 . . . . . 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 477 . . . 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 10764 . . 3 |- ( ( ( ( A - B ) e. QQ /\ B e. QQ ) /\ ( 0 <_ ( A - B ) /\ ( A - B ) < B ) ) -> ( ( A - B ) mod B ) = ( A - B ) )
2914, 9, 27, 28syl21anc 1277 . 2 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( ( A - B ) mod B ) = ( A - B ) )
306, 12, 293eqtr3d 2279 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 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   < clt 8350   <_ cle 8351   - cmin 8487   2c2 9334   ZZcz 9623   QQcq 9998   mod cmo 10737
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288
This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-q 9999  df-rp 10034  df-fl 10683  df-mod 10738
This theorem is referenced by:  modifeq2int  10801  modaddmodup  10802
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