Theorem modq0 10563

Description: A mod B is zero iff A is evenly divisible by B. (Contributed by Jim Kingdon, 17-Oct-2021.)

Assertion

Ref	Expression
modq0	|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( A / B ) e. ZZ ) )

Proof of Theorem modq0

Step	Hyp	Ref	Expression
1		modqval 10558	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
2	1	eqeq1d 2238	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( A - ( B x. ( |_ ` ( A / B ) ) ) ) = 0 ) )
3		qcn 9841	. . . . . 6 |- ( A e. QQ -> A e. CC )
4	3	3ad2ant1 1042	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> A e. CC )
5		qcn 9841	. . . . . . 7 |- ( B e. QQ -> B e. CC )
6	5	3ad2ant2 1043	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. CC )
7		simp3 1023	. . . . . . . . . 10 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 < B )
8	7	gt0ne0d 8670	. . . . . . . . 9 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B =/= 0 )
9		qdivcl 9850	. . . . . . . . 9 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
10	8, 9	syld3an3 1316	. . . . . . . 8 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A / B ) e. QQ )
11	10	flqcld 10509	. . . . . . 7 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. ZZ )
12	11	zcnd 9581	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( |_ ` ( A / B ) ) e. CC )
13	6, 12	mulcld 8178	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( B x. ( |_ ` ( A / B ) ) ) e. CC )
14	4, 13	subeq0ad 8478	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A - ( B x. ( |_ ` ( A / B ) ) ) ) = 0 <-> A = ( B x. ( |_ ` ( A / B ) ) ) ) )
15	2, 14	bitrd 188	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> A = ( B x. ( |_ ` ( A / B ) ) ) ) )
16		qre 9832	. . . . . . 7 |- ( B e. QQ -> B e. RR )
17	16	3ad2ant2 1043	. . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B e. RR )
18	17, 7	gt0ap0d 8787	. . . . 5 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> B # 0 )
19	4, 12, 6, 18	divmulap2d 8982	. . . 4 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A / B ) = ( |_ ` ( A / B ) ) <-> A = ( B x. ( |_ ` ( A / B ) ) ) ) )
20		eqcom 2231	. . . 4 |- ( ( A / B ) = ( |_ ` ( A / B ) ) <-> ( |_ ` ( A / B ) ) = ( A / B ) )
21	19, 20	bitr3di 195	. . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A = ( B x. ( |_ ` ( A / B ) ) ) <-> ( |_ ` ( A / B ) ) = ( A / B ) ) )
22	15, 21	bitrd 188	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( |_ ` ( A / B ) ) = ( A / B ) ) )
23		flqidz 10518	. . 3 |- ( ( A / B ) e. QQ -> ( ( |_ ` ( A / B ) ) = ( A / B ) <-> ( A / B ) e. ZZ ) )
24	10, 23	syl 14	. 2 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( |_ ` ( A / B ) ) = ( A / B ) <-> ( A / B ) e. ZZ ) )
25	22, 24	bitrd 188	1 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( A / B ) e. ZZ ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   CCcc 8008   RRcr 8009   0cc0 8010   x. cmul 8015   < clt 8192   - cmin 8328   / cdiv 8830   ZZcz 9457   QQcq 9826   |_cfl 10500   mod cmo 10556

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-n0 9381  df-z 9458  df-q 9827  df-rp 9862  df-fl 10502  df-mod 10557

This theorem is referenced by:  mulqmod0  10564  negqmod0  10565  modqid0  10584  q2txmodxeq0  10618  addmodlteq  10632  dvdsval3  12317