Theorem lediv1 9016

Description: Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)

Assertion

Ref	Expression
lediv1	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( A / C ) <_ ( B / C ) ) )

Proof of Theorem lediv1

Step	Hyp	Ref	Expression
1		ltdiv1 9015	. . . 4 |- ( ( B e. RR /\ A e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B < A <-> ( B / C ) < ( A / C ) ) )
2	1	3com12 1231	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B < A <-> ( B / C ) < ( A / C ) ) )
3	2	notbid 671	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( -. B < A <-> -. ( B / C ) < ( A / C ) ) )
4		lenlt 8222	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )
5	4	3adant3 1041	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> -. B < A ) )
6		gt0ap0 8773	. . . . . . 7 |- ( ( C e. RR /\ 0 < C ) -> C # 0 )
7	6	3adant1 1039	. . . . . 6 |- ( ( A e. RR /\ C e. RR /\ 0 < C ) -> C # 0 )
8		redivclap 8878	. . . . . 6 |- ( ( A e. RR /\ C e. RR /\ C # 0 ) -> ( A / C ) e. RR )
9	7, 8	syld3an3 1316	. . . . 5 |- ( ( A e. RR /\ C e. RR /\ 0 < C ) -> ( A / C ) e. RR )
10	9	3expb 1228	. . . 4 |- ( ( A e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A / C ) e. RR )
11	10	3adant2 1040	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A / C ) e. RR )
12	6	3adant1 1039	. . . . . 6 |- ( ( B e. RR /\ C e. RR /\ 0 < C ) -> C # 0 )
13		redivclap 8878	. . . . . 6 |- ( ( B e. RR /\ C e. RR /\ C # 0 ) -> ( B / C ) e. RR )
14	12, 13	syld3an3 1316	. . . . 5 |- ( ( B e. RR /\ C e. RR /\ 0 < C ) -> ( B / C ) e. RR )
15	14	3expb 1228	. . . 4 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B / C ) e. RR )
16	15	3adant1 1039	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B / C ) e. RR )
17	11, 16	lenltd 8264	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) <_ ( B / C ) <-> -. ( B / C ) < ( A / C ) ) )
18	3, 5, 17	3bitr4d 220	1 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( A / C ) <_ ( B / C ) ) )

Syntax hints:   -. wn 3   -> 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   < clt 8181   <_ cle 8182   # cap 8728   / 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:  ge0div  9018  ledivmul  9024  lediv23  9040  lediv1d  9939  icccntr  10196  sin01bnd  12268  cos01bnd  12269  sin02gt0  12275  hashdvds  12743  cosordlem  15523  gausslemma2dlem1a  15737  gausslemma2dlem3  15742  lgseisenlem1  15749  2lgslem1c  15769