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Theorem ledivmul2OLD 9893
Description: 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ledivmul2OLD  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  ( ( A  /  B )  <_  C 
<->  A  <_  ( C  x.  B ) ) )

Proof of Theorem ledivmul2OLD
StepHypRef Expression
1 simpl1 961 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  A  e.  RR )
2 simpl3 963 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  C  e.  RR )
3 simp2 959 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
4 id 21 . . . 4  |-  ( 0  <  B  ->  0  <  B )
53, 4anim12i 551 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  ( B  e.  RR  /\  0  < 
B ) )
6 ledivmul 9888 . . 3  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( A  /  B )  <_  C  <->  A  <_  ( B  x.  C ) ) )
71, 2, 5, 6syl3anc 1185 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  ( ( A  /  B )  <_  C 
<->  A  <_  ( B  x.  C ) ) )
8 recn 9085 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
9 recn 9085 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
10 mulcom 9081 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
118, 9, 10syl2an 465 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
12113adant1 976 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C )  =  ( C  x.  B ) )
1312adantr 453 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  ( B  x.  C )  =  ( C  x.  B ) )
1413breq2d 4227 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  ( A  <_ 
( B  x.  C
)  <->  A  <_  ( C  x.  B ) ) )
157, 14bitrd 246 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  ( ( A  /  B )  <_  C 
<->  A  <_  ( C  x.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215  (class class class)co 6084   CCcc 8993   RRcr 8994   0cc0 8995    x. cmul 9000    < clt 9125    <_ cle 9126    / cdiv 9682
This theorem is referenced by:  nmblolbii  22305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683
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