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Theorem ledivmul2OLD 10412
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 994 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  A  e.  RR )
2 simpl3 996 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  C  e.  RR )
3 simp2 992 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
4 id 22 . . . 4  |-  ( 0  <  B  ->  0  <  B )
53, 4anim12i 566 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  ( B  e.  RR  /\  0  < 
B ) )
6 ledivmul 10407 . . 3  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( A  /  B )  <_  C  <->  A  <_  ( B  x.  C ) ) )
71, 2, 5, 6syl3anc 1223 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  ( ( A  /  B )  <_  C 
<->  A  <_  ( B  x.  C ) ) )
8 recn 9571 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
9 recn 9571 . . . . . 6  |-  ( C  e.  RR  ->  C  e.  CC )
10 mulcom 9567 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
118, 9, 10syl2an 477 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
12113adant1 1009 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C )  =  ( C  x.  B ) )
1312adantr 465 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  ( B  x.  C )  =  ( C  x.  B ) )
1413breq2d 4452 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  ( A  <_ 
( B  x.  C
)  <->  A  <_  ( C  x.  B ) ) )
157, 14bitrd 253 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  0  <  B )  ->  ( ( A  /  B )  <_  C 
<->  A  <_  ( C  x.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4440  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486    < clt 9617    <_ cle 9618    / cdiv 10195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196
This theorem is referenced by:  nmblolbii  25376
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