Theorem ltdivmul 9019

Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Assertion

Ref	Expression
ltdivmul	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) < B <-> A < ( C x. B ) ) )

Proof of Theorem ltdivmul

Step	Hyp	Ref	Expression
1		remulcl 8123	. . . . . 6 |- ( ( C e. RR /\ B e. RR ) -> ( C x. B ) e. RR )
2	1	ancoms 268	. . . . 5 |- ( ( B e. RR /\ C e. RR ) -> ( C x. B ) e. RR )
3	2	adantrr 479	. . . 4 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
4	3	3adant1 1039	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
5		ltdiv1 9011	. . 3 |- ( ( A e. RR /\ ( C x. B ) e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < ( ( C x. B ) / C ) ) )
6	4, 5	syld3an2 1318	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < ( ( C x. B ) / C ) ) )
7		recn 8128	. . . . . 6 |- ( B e. RR -> B e. CC )
8	7	adantr 276	. . . . 5 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> B e. CC )
9		recn 8128	. . . . . 6 |- ( C e. RR -> C e. CC )
10	9	ad2antrl 490	. . . . 5 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> C e. CC )
11		gt0ap0 8769	. . . . . 6 |- ( ( C e. RR /\ 0 < C ) -> C # 0 )
12	11	adantl 277	. . . . 5 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> C # 0 )
13	8, 10, 12	divcanap3d 8938	. . . 4 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
14	13	3adant1 1039	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
15	14	breq2d 4094	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) < ( ( C x. B ) / C ) <-> ( A / C ) < B ) )
16	6, 15	bitr2d 189	1 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) < B <-> A < ( C x. B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4082  (class class class)co 6000   CCcc 7993   RRcr 7994   0cc0 7995   x. cmul 8000   < clt 8177   # cap 8724   / cdiv 8815

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113

This theorem depends on definitions:  df-bi 117  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-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 3888  df-br 4083  df-opab 4145  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816

This theorem is referenced by:  ltdivmul2  9021  lt2mul2div  9022  ltrec  9026  avglt2  9347  3halfnz  9540  ltdivmuld  9940  modqid  10566  expnbnd  10880  mertenslemi1  12041  eirraplem  12283  fldivp1  12866  pcfaclem  12867  4sqlem12  12920  dveflem  15394  coseq0negpitopi  15504  tangtx  15506  cosordlem  15517  cos02pilt1  15519  gausslemma2dlem0c  15724  lgsquadlem1  15750  2sqlem8  15796  ex-fl  16047