Theorem lediv1 9199

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 9198	. . . 4 |- ( ( B e. RR /\ A e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B < A <-> ( B / C ) < ( A / C ) ) )
2	1	3com12 1238	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B < A <-> ( B / C ) < ( A / C ) ) )
3	2	notbid 677	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( -. B < A <-> -. ( B / C ) < ( A / C ) ) )
4		lenlt 8401	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )
5	4	3adant3 1048	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> -. B < A ) )
6		gt0ap0 8954	. . . . . . 7 |- ( ( C e. RR /\ 0 < C ) -> C # 0 )
7	6	3adant1 1046	. . . . . 6 |- ( ( A e. RR /\ C e. RR /\ 0 < C ) -> C # 0 )
8		redivclap 9061	. . . . . 6 |- ( ( A e. RR /\ C e. RR /\ C # 0 ) -> ( A / C ) e. RR )
9	7, 8	syld3an3 1323	. . . . 5 |- ( ( A e. RR /\ C e. RR /\ 0 < C ) -> ( A / C ) e. RR )
10	9	3expb 1235	. . . 4 |- ( ( A e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A / C ) e. RR )
11	10	3adant2 1047	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A / C ) e. RR )
12	6	3adant1 1046	. . . . . 6 |- ( ( B e. RR /\ C e. RR /\ 0 < C ) -> C # 0 )
13		redivclap 9061	. . . . . 6 |- ( ( B e. RR /\ C e. RR /\ C # 0 ) -> ( B / C ) e. RR )
14	12, 13	syld3an3 1323	. . . . 5 |- ( ( B e. RR /\ C e. RR /\ 0 < C ) -> ( B / C ) e. RR )
15	14	3expb 1235	. . . 4 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B / C ) e. RR )
16	15	3adant1 1046	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B / C ) e. RR )
17	11, 16	lenltd 8444	. 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 ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   RRcr 8178   0cc0 8179   < clt 8360   <_ cle 8361   # 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:  ge0div  9201  ledivmul  9207  lediv23  9223  lediv1d  10144  icccntr  10402  sin01bnd  12524  cos01bnd  12525  sin02gt0  12531  hashdvds  12999  cosordlem  15950  gausslemma2dlem1a  16177  gausslemma2dlem3  16182  lgseisenlem1  16189  2lgslem1c  16209