Theorem ltmuldiv 9021

Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

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

Proof of Theorem ltmuldiv

Step	Hyp	Ref	Expression
1		simp1 1021	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> A e. RR )
2		simp3l 1049	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> C e. RR )
3	1, 2	remulcld 8177	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A x. C ) e. RR )
4		ltdiv1 9015	. . 3 |- ( ( ( A x. C ) e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) < B <-> ( ( A x. C ) / C ) < ( B / C ) ) )
5	3, 4	syld3an1 1317	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) < B <-> ( ( A x. C ) / C ) < ( B / C ) ) )
6	1	recnd 8175	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> A e. CC )
7	2	recnd 8175	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> C e. CC )
8		simp3r 1050	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> 0 < C )
9	2, 8	gt0ap0d 8776	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> C # 0 )
10	6, 7, 9	divcanap4d 8943	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) / C ) = A )
11	10	breq1d 4093	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( ( A x. C ) / C ) < ( B / C ) <-> A < ( B / C ) ) )
12	5, 11	bitrd 188	1 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) < B <-> A < ( B / C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   RRcr 7998   0cc0 7999   x. cmul 8004   < clt 8181   / cdiv 8819

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 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

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 3889  df-br 4084  df-opab 4146  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-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820

This theorem is referenced by:  ltmuldiv2  9022  lt2mul2div  9026  ltrec  9030  ltmuldivi  9069  avglt1  9350  3halfnz  9544  ltmuldivd  9940  nno  12417  prmind2  12642  hashgcdlem  12760  reeff1oleme  15446  tangtx  15512  logdivlti  15555  mersenne  15671