Theorem lediv1 8764

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 8763	. . . 4 |- ( ( B e. RR /\ A e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B < A <-> ( B / C ) < ( A / C ) ) )
2	1	3com12 1197	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B < A <-> ( B / C ) < ( A / C ) ) )
3	2	notbid 657	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( -. B < A <-> -. ( B / C ) < ( A / C ) ) )
4		lenlt 7974	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )
5	4	3adant3 1007	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> -. B < A ) )
6		gt0ap0 8524	. . . . . . 7 |- ( ( C e. RR /\ 0 < C ) -> C # 0 )
7	6	3adant1 1005	. . . . . 6 |- ( ( A e. RR /\ C e. RR /\ 0 < C ) -> C # 0 )
8		redivclap 8627	. . . . . 6 |- ( ( A e. RR /\ C e. RR /\ C # 0 ) -> ( A / C ) e. RR )
9	7, 8	syld3an3 1273	. . . . 5 |- ( ( A e. RR /\ C e. RR /\ 0 < C ) -> ( A / C ) e. RR )
10	9	3expb 1194	. . . 4 |- ( ( A e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A / C ) e. RR )
11	10	3adant2 1006	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A / C ) e. RR )
12	6	3adant1 1005	. . . . . 6 |- ( ( B e. RR /\ C e. RR /\ 0 < C ) -> C # 0 )
13		redivclap 8627	. . . . . 6 |- ( ( B e. RR /\ C e. RR /\ C # 0 ) -> ( B / C ) e. RR )
14	12, 13	syld3an3 1273	. . . . 5 |- ( ( B e. RR /\ C e. RR /\ 0 < C ) -> ( B / C ) e. RR )
15	14	3expb 1194	. . . 4 |- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B / C ) e. RR )
16	15	3adant1 1005	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( B / C ) e. RR )
17	11, 16	lenltd 8016	. 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 219	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 103   <-> wb 104   /\ w3a 968   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   < clt 7933   <_ cle 7934   # cap 8479   / cdiv 8568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569

This theorem is referenced by:  ge0div  8766  ledivmul  8772  lediv23  8788  lediv1d  9679  icccntr  9936  sin01bnd  11698  cos01bnd  11699  sin02gt0  11704  hashdvds  12153  cosordlem  13410