Theorem q2submod 10151

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

Step	Hyp	Ref	Expression
1		qcn 9419	. . . . . . 7 |- ( B e. QQ -> B e. CC )
2	1	3ad2ant2 1003	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. CC )
3	2	adantr 274	. . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> B e. CC )
4	3	mulid1d 7776	. . . 4 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( B x. 1 ) = B )
5	4	oveq2d 5783	. . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A - ( B x. 1 ) ) = ( A - B ) )
6	5	oveq1d 5782	. 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 984	. . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> A e. QQ )
8		1zzd 9074	. . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> 1 e. ZZ )
9		simpl2 985	. . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> B e. QQ )
10		simpl3 986	. . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> 0 < B )
11		modqcyc2 10126	. . 3 |- ( ( ( A e. QQ /\ 1 e. ZZ ) /\ ( B e. QQ /\ 0 < B ) ) -> ( ( A - ( B x. 1 ) ) mod B ) = ( A mod B ) )
12	7, 8, 9, 10, 11	syl22anc 1217	. 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 9423	. . . 4 |- ( ( A e. QQ /\ B e. QQ ) -> ( A - B ) e. QQ )
14	7, 9, 13	syl2anc 408	. . 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 9410	. . . . . . . 8 |- ( A e. QQ -> A e. RR )
17	7, 16	syl 14	. . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> A e. RR )
18		qre 9410	. . . . . . . 8 |- ( B e. QQ -> B e. RR )
19	9, 18	syl 14	. . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> B e. RR )
20	17, 19	subge0d 8290	. . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( 0 <_ ( A - B ) <-> B <_ A ) )
21	20	bicomd 140	. . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( B <_ A <-> 0 <_ ( A - B ) ) )
22	3	2timesd 8955	. . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( 2 x. B ) = ( B + B ) )
23	22	breq2d 3936	. . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A < ( 2 x. B ) <-> A < ( B + B ) ) )
24	17, 19, 19	ltsubaddd 8296	. . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( ( A - B ) < B <-> A < ( B + B ) ) )
25	23, 24	bitr4d 190	. . . . 5 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A < ( 2 x. B ) <-> ( A - B ) < B ) )
26	21, 25	anbi12d 464	. . . 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 ) ) )
27	15, 26	mpbid 146	. . 3 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( 0 <_ ( A - B ) /\ ( A - B ) < B ) )
28		modqid 10115	. . 3 |- ( ( ( ( A - B ) e. QQ /\ B e. QQ ) /\ ( 0 <_ ( A - B ) /\ ( A - B ) < B ) ) -> ( ( A - B ) mod B ) = ( A - B ) )
29	14, 9, 27, 28	syl21anc 1215	. 2 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( ( A - B ) mod B ) = ( A - B ) )
30	6, 12, 29	3eqtr3d 2178	1 |- ( ( ( A e. QQ /\ B e. QQ /\ 0 < B ) /\ ( B <_ A /\ A < ( 2 x. B ) ) ) -> ( A mod B ) = ( A - B ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3924  (class class class)co 5767   CCcc 7611   RRcr 7612   0cc0 7613   1c1 7614   + caddc 7616   x. cmul 7618   < clt 7793   <_ cle 7794   - cmin 7926   2c2 8764   ZZcz 9047   QQcq 9404   mod cmo 10088

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-n0 8971  df-z 9048  df-q 9405  df-rp 9435  df-fl 10036  df-mod 10089

This theorem is referenced by:  modifeq2int  10152  modaddmodup  10153