Theorem ltdivmul 9206

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 8307	. . . . . 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 483	. . . 4 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
4	3	3adant1 1046	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
5		ltdiv1 9198	. . 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 1325	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < ( ( C x. B ) / C ) ) )
7		recn 8312	. . . . . 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 8312	. . . . . 6 |- ( C e. RR -> C e. CC )
10	9	ad2antrl 494	. . . . 5 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> C e. CC )
11		gt0ap0 8954	. . . . . 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 9125	. . . 4 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
14	13	3adant1 1046	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
15	14	breq2d 4142	. 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   x. cmul 8184   < clt 8360   # cap 8909   / cdiv 9002

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  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-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003

This theorem is used by:  ltdivmul2  9208  lt2mul2div  9209  ltrec  9213  avglt2  9545  3halfnz  9743  ltdivmuld  10149  modqid  10786  expnbnd  11101  mertenslemi1  12302  eirraplem  12544  fldivp1  13127  pcfaclem  13128  4sqlem12  13181  dveflem  15827  coseq0negpitopi  15937  tangtx  15939  cosordlem  15950  cos02pilt1  15952  birthdaylem3  16089  gausslemma2dlem0c  16170  lgsquadlem1  16196  2sqlem8  16242  ex-fl  16739