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Theorem unitdivcld 26343
Description: Necessary conditions for a quotient to be in the closed unit. (somewhat too strong, it would be sufficient that A and B are in RR+) (Contributed by Thierry Arnoux, 20-Dec-2016.)
Assertion
Ref Expression
unitdivcld  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  <_  B  <->  ( A  /  B )  e.  ( 0 [,] 1 ) ) )

Proof of Theorem unitdivcld
StepHypRef Expression
1 elunitrn 26339 . . . . . . . 8  |-  ( A  e.  ( 0 [,] 1 )  ->  A  e.  RR )
213ad2ant1 1009 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  ->  A  e.  RR )
3 elunitrn 26339 . . . . . . . 8  |-  ( B  e.  ( 0 [,] 1 )  ->  B  e.  RR )
433ad2ant2 1010 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  ->  B  e.  RR )
5 simp3 990 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  ->  B  =/=  0 )
62, 4, 5redivcld 10171 . . . . . 6  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  /  B
)  e.  RR )
76adantr 465 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0
)  /\  A  <_  B )  ->  ( A  /  B )  e.  RR )
8 elunitge0 26341 . . . . . . . 8  |-  ( A  e.  ( 0 [,] 1 )  ->  0  <_  A )
983ad2ant1 1009 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <_  A )
10 elunitge0 26341 . . . . . . . . . 10  |-  ( B  e.  ( 0 [,] 1 )  ->  0  <_  B )
1110adantr 465 . . . . . . . . 9  |-  ( ( B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <_  B )
12 0re 9398 . . . . . . . . . . . . 13  |-  0  e.  RR
13 ltlen 9488 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
1412, 3, 13sylancr 663 . . . . . . . . . . . 12  |-  ( B  e.  ( 0 [,] 1 )  ->  (
0  <  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
1514biimpar 485 . . . . . . . . . . 11  |-  ( ( B  e.  ( 0 [,] 1 )  /\  ( 0  <_  B  /\  B  =/=  0
) )  ->  0  <  B )
16153impb 1183 . . . . . . . . . 10  |-  ( ( B  e.  ( 0 [,] 1 )  /\  0  <_  B  /\  B  =/=  0 )  ->  0  <  B )
17163com23 1193 . . . . . . . . 9  |-  ( ( B  e.  ( 0 [,] 1 )  /\  B  =/=  0  /\  0  <_  B )  ->  0  <  B )
1811, 17mpd3an3 1315 . . . . . . . 8  |-  ( ( B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <  B )
19183adant1 1006 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <  B )
20 divge0 10210 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  0  <_  ( A  /  B ) )
212, 9, 4, 19, 20syl22anc 1219 . . . . . 6  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <_  ( A  /  B ) )
2221adantr 465 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0
)  /\  A  <_  B )  ->  0  <_  ( A  /  B ) )
23 1re 9397 . . . . . . . . 9  |-  1  e.  RR
2423a1i 11 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
1  e.  RR )
25 ledivmul 10217 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( A  /  B )  <_  1  <->  A  <_  ( B  x.  1 ) ) )
262, 24, 4, 19, 25syl112anc 1222 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( ( A  /  B )  <_  1  <->  A  <_  ( B  x.  1 ) ) )
27 ax-1rid 9364 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
2827breq2d 4316 . . . . . . . 8  |-  ( B  e.  RR  ->  ( A  <_  ( B  x.  1 )  <->  A  <_  B ) )
294, 28syl 16 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  <_  ( B  x.  1 )  <-> 
A  <_  B )
)
3026, 29bitr2d 254 . . . . . 6  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  <_  B  <->  ( A  /  B )  <_  1 ) )
3130biimpa 484 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0
)  /\  A  <_  B )  ->  ( A  /  B )  <_  1
)
327, 22, 313jca 1168 . . . 4  |-  ( ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0
)  /\  A  <_  B )  ->  ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
)  /\  ( A  /  B )  <_  1
) )
3332ex 434 . . 3  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  <_  B  ->  ( ( A  /  B )  e.  RR  /\  0  <_  ( A  /  B )  /\  ( A  /  B )  <_ 
1 ) ) )
34 simp3 990 . . . 4  |-  ( ( ( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B )  /\  ( A  /  B )  <_ 
1 )  ->  ( A  /  B )  <_ 
1 )
3534, 30syl5ibr 221 . . 3  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
)  /\  ( A  /  B )  <_  1
)  ->  A  <_  B ) )
3633, 35impbid 191 . 2  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  <_  B  <->  ( ( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B )  /\  ( A  /  B )  <_ 
1 ) ) )
3712, 23elicc2i 11373 . 2  |-  ( ( A  /  B )  e.  ( 0 [,] 1 )  <->  ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
)  /\  ( A  /  B )  <_  1
) )
3836, 37syl6bbr 263 1  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  <_  B  <->  ( A  /  B )  e.  ( 0 [,] 1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1756    =/= wne 2618   class class class wbr 4304  (class class class)co 6103   RRcr 9293   0cc0 9294   1c1 9295    x. cmul 9299    < clt 9430    <_ cle 9431    / cdiv 10005   [,]cicc 11315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-icc 11319
This theorem is referenced by:  cndprob01  26830
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