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Theorem ltdiv2 10491
Description: Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)
Assertion
Ref Expression
ltdiv2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( C  /  B )  <  ( C  /  A ) ) )

Proof of Theorem ltdiv2
StepHypRef Expression
1 ltrec 10487 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  <  B  <->  ( 1  /  B )  <  ( 1  /  A ) ) )
213adant3 1025 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( 1  /  B )  <  ( 1  /  A ) ) )
3 gt0ne0 10078 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
4 rereccl 10324 . . . . . 6  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  RR )
53, 4syldan 472 . . . . 5  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( 1  /  B
)  e.  RR )
6 gt0ne0 10078 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
7 rereccl 10324 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  RR )
86, 7syldan 472 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
9 ltmul2 10455 . . . . . 6  |-  ( ( ( 1  /  B
)  e.  RR  /\  ( 1  /  A
)  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( (
1  /  B )  <  ( 1  /  A )  <->  ( C  x.  ( 1  /  B
) )  <  ( C  x.  ( 1  /  A ) ) ) )
108, 9syl3an2 1298 . . . . 5  |-  ( ( ( 1  /  B
)  e.  RR  /\  ( A  e.  RR  /\  0  <  A )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( 1  /  B )  <  (
1  /  A )  <-> 
( C  x.  (
1  /  B ) )  <  ( C  x.  ( 1  /  A ) ) ) )
115, 10syl3an1 1297 . . . 4  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( A  e.  RR  /\  0  < 
A )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( 1  /  B )  <  (
1  /  A )  <-> 
( C  x.  (
1  /  B ) )  <  ( C  x.  ( 1  /  A ) ) ) )
12 recn 9628 . . . . . . 7  |-  ( C  e.  RR  ->  C  e.  CC )
1312adantr 466 . . . . . 6  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  e.  CC )
14 recn 9628 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  CC )
1514adantr 466 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  CC )
1615, 3jca 534 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
17 recn 9628 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
1817adantr 466 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
1918, 6jca 534 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  e.  CC  /\  A  =/=  0 ) )
20 divrec 10285 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( C  /  B )  =  ( C  x.  (
1  /  B ) ) )
21203expb 1206 . . . . . . . 8  |-  ( ( C  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( C  /  B )  =  ( C  x.  ( 1  /  B ) ) )
22213adant3 1025 . . . . . . 7  |-  ( ( C  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( C  /  B
)  =  ( C  x.  ( 1  /  B ) ) )
23 divrec 10285 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  ->  ( C  /  A )  =  ( C  x.  (
1  /  A ) ) )
24233expb 1206 . . . . . . . 8  |-  ( ( C  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) )  ->  ( C  /  A )  =  ( C  x.  ( 1  /  A ) ) )
25243adant2 1024 . . . . . . 7  |-  ( ( C  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( C  /  A
)  =  ( C  x.  ( 1  /  A ) ) )
2622, 25breq12d 4439 . . . . . 6  |-  ( ( C  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( C  /  B )  <  ( C  /  A )  <->  ( C  x.  ( 1  /  B
) )  <  ( C  x.  ( 1  /  A ) ) ) )
2713, 16, 19, 26syl3an 1306 . . . . 5  |-  ( ( ( C  e.  RR  /\  0  <  C )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( C  /  B )  <  ( C  /  A )  <->  ( C  x.  ( 1  /  B
) )  <  ( C  x.  ( 1  /  A ) ) ) )
28273coml 1212 . . . 4  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( A  e.  RR  /\  0  < 
A )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( C  /  B )  <  ( C  /  A )  <->  ( C  x.  ( 1  /  B
) )  <  ( C  x.  ( 1  /  A ) ) ) )
2911, 28bitr4d 259 . . 3  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( A  e.  RR  /\  0  < 
A )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( 1  /  B )  <  (
1  /  A )  <-> 
( C  /  B
)  <  ( C  /  A ) ) )
30293com12 1209 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( 1  /  B )  <  (
1  /  A )  <-> 
( C  /  B
)  <  ( C  /  A ) ) )
312, 30bitrd 256 1  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( C  /  B )  <  ( C  /  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543    < clt 9674    / cdiv 10268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269
This theorem is referenced by:  ltdiv2OLD  10492  ltdiv2d  11364  perfectlem2  24021  bposlem6  24080  dchrisum0flblem2  24210  pntpbnd1a  24286  pntlemr  24303  stoweidlem42  37471
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