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Theorem lediv23 10505
Description: Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
Assertion
Ref Expression
lediv23  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  <_  C  <->  ( A  /  C )  <_  B ) )

Proof of Theorem lediv23
StepHypRef Expression
1 simpl 459 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  RR )
2 gt0ne0 10086 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
31, 2jca 535 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  RR  /\  B  =/=  0 ) )
4 redivcl 10333 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  /  B )  e.  RR )
543expb 1210 . . . . . 6  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( A  /  B )  e.  RR )
63, 5sylan2 477 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  /  B )  e.  RR )
763adant3 1029 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( A  /  B )  e.  RR )
8 simp3 1011 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  C  e.  RR )
9 simp2 1010 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( B  e.  RR  /\  0  < 
B ) )
10 lemul1 10464 . . . 4  |-  ( ( ( A  /  B
)  e.  RR  /\  C  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( A  /  B )  <_  C  <->  ( ( A  /  B
)  x.  B )  <_  ( C  x.  B ) ) )
117, 8, 9, 10syl3anc 1269 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( ( A  /  B )  <_  C 
<->  ( ( A  /  B )  x.  B
)  <_  ( C  x.  B ) ) )
12113adant3r 1266 . 2  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  <_  C  <->  ( ( A  /  B
)  x.  B )  <_  ( C  x.  B ) ) )
13 recn 9634 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
1413adantr 467 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  A  e.  CC )
15 recn 9634 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
1615ad2antrl 735 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  e.  CC )
172adantl 468 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  =/=  0 )
1814, 16, 17divcan1d 10391 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  x.  B )  =  A )
19183adant3 1029 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  x.  B
)  =  A )
2019breq1d 4415 . 2  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( ( A  /  B )  x.  B )  <_  ( C  x.  B )  <->  A  <_  ( C  x.  B ) ) )
21 remulcl 9629 . . . . . . . 8  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  x.  B
)  e.  RR )
2221ancoms 455 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  x.  B
)  e.  RR )
2322adantrr 724 . . . . . 6  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( C  x.  B )  e.  RR )
24233adant1 1027 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( C  x.  B
)  e.  RR )
25 lediv1 10477 . . . . 5  |-  ( ( A  e.  RR  /\  ( C  x.  B
)  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <_  ( C  x.  B
)  <->  ( A  /  C )  <_  (
( C  x.  B
)  /  C ) ) )
2624, 25syld3an2 1316 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  ( C  x.  B )  <->  ( A  /  C )  <_  ( ( C  x.  B )  /  C ) ) )
27 recn 9634 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  CC )
2827adantr 467 . . . . . . . 8  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  e.  CC )
29 gt0ne0 10086 . . . . . . . 8  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
3028, 29jca 535 . . . . . . 7  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
31 divcan3 10301 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  (
( C  x.  B
)  /  C )  =  B )
32313expb 1210 . . . . . . 7  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( C  x.  B )  /  C )  =  B )
3315, 30, 32syl2an 480 . . . . . 6  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( C  x.  B )  /  C )  =  B )
34333adant1 1027 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( C  x.  B )  /  C
)  =  B )
3534breq2d 4417 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <_  (
( C  x.  B
)  /  C )  <-> 
( A  /  C
)  <_  B )
)
3626, 35bitrd 257 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  ( C  x.  B )  <->  ( A  /  C )  <_  B ) )
37363adant2r 1264 . 2  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( A  <_  ( C  x.  B )  <->  ( A  /  C )  <_  B ) )
3812, 20, 373bitrd 283 1  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  <_  C  <->  ( A  /  C )  <_  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   class class class wbr 4405  (class class class)co 6295   CCcc 9542   RRcr 9543   0cc0 9544    x. cmul 9549    < clt 9680    <_ cle 9681    / cdiv 10276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277
This theorem is referenced by:  lediv23d  11411  pntlemj  24453  minvecolem4  26534  minvecolem4OLD  26544  stoweidlem36  37907
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