Theorem modqlt 10719

Description: The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.)

Assertion

Ref	Expression
modqlt	|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) < B )

Proof of Theorem modqlt

Step	Hyp	Ref	Expression
1		qcn 9984	. . . . . 6 |- ( A e. QQ -> A e. CC )
2	1	3ad2ant1 1045	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> A e. CC )
3		qcn 9984	. . . . . 6 |- ( B e. QQ -> B e. CC )
4	3	3ad2ant2 1046	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. CC )
5		qre 9975	. . . . . . 7 |- ( B e. QQ -> B e. RR )
6	5	3ad2ant2 1046	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR )
7		simp3 1026	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 < B )
8	6, 7	gt0ap0d 8920	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B # 0 )
9	2, 4, 8	divcanap2d 9083	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( A / B ) ) = A )
10	9	oveq1d 6073	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( B x. ( A / B ) ) - ( B x. ( |_ ` ( A / B ) ) ) ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
11	7	gt0ne0d 8803	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B =/= 0 )
12		qdivcl 9993	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
13	11, 12	syld3an3 1319	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. QQ )
14		qcn 9984	. . . . 5 |- ( ( A / B ) e. QQ -> ( A / B ) e. CC )
15	13, 14	syl 14	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. CC )
16	13	flqcld 10661	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. ZZ )
17	16	zcnd 9719	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. CC )
18	4, 15, 17	subdid 8704	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) = ( ( B x. ( A / B ) ) - ( B x. ( |_ ` ( A / B ) ) ) ) )
19		modqval 10710	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
20	10, 18, 19	3eqtr4rd 2278	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) )
21		qfraclt1 10664	. . . . 5 |- ( ( A / B ) e. QQ -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < 1 )
22	13, 21	syl 14	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < 1 )
23	4, 8	dividapd 9077	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B / B ) = 1 )
24	22, 23	breqtrrd 4142	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < ( B / B ) )
25		qre 9975	. . . . . 6 |- ( ( A / B ) e. QQ -> ( A / B ) e. RR )
26	13, 25	syl 14	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. RR )
27	16	zred 9718	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. RR )
28	26, 27	resubcld 8671	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) e. RR )
29		ltmuldiv2 9166	. . . 4 |- ( ( ( ( A / B ) - ( |_ ` ( A / B ) ) ) e. RR /\ B e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) < B <-> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < ( B / B ) ) )
30	28, 6, 6, 7, 29	syl112anc 1278	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) < B <-> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < ( B / B ) ) )
31	24, 30	mpbird 167	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( ( A / B ) - ( |_ ` ( A / B ) ) ) ) < B )
32	20, 31	eqbrtrd 4136	1 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) < B )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   e. wcel 2205   =/= wne 2414   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   x. cmul 8148   < clt 8324   - cmin 8460   / cdiv 8963   QQcq 9969   |_cfl 10652   mod cmo 10708

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-n0 9514  df-z 9595  df-q 9970  df-rp 10005  df-fl 10654  df-mod 10709

This theorem is referenced by:  modqelico  10720  zmodfz  10732  modqid2  10737  modqabs  10743  modqmuladdim  10753  modaddmodup  10773  modqsubdir  10779  divalglemnn  12629  divalgmod  12638  bitsmod  12667  bitsinv1lem  12672  bezoutlemnewy  12717  bezoutlemstep  12718  eucalglt  12779  odzdvds  12968  fldivp1  13071  4sqlem6  13106  4sqlem12  13125  lgseisenlem1  16069