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Theorem unitdivcld 28719
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 28715 . . . . . . . 8  |-  ( A  e.  ( 0 [,] 1 )  ->  A  e.  RR )
213ad2ant1 1030 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  ->  A  e.  RR )
3 elunitrn 28715 . . . . . . . 8  |-  ( B  e.  ( 0 [,] 1 )  ->  B  e.  RR )
433ad2ant2 1031 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  ->  B  e.  RR )
5 simp3 1011 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  ->  B  =/=  0 )
62, 4, 5redivcld 10442 . . . . . 6  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  /  B
)  e.  RR )
76adantr 467 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0
)  /\  A  <_  B )  ->  ( A  /  B )  e.  RR )
8 elunitge0 28717 . . . . . . . 8  |-  ( A  e.  ( 0 [,] 1 )  ->  0  <_  A )
983ad2ant1 1030 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <_  A )
10 elunitge0 28717 . . . . . . . . . 10  |-  ( B  e.  ( 0 [,] 1 )  ->  0  <_  B )
1110adantr 467 . . . . . . . . 9  |-  ( ( B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <_  B )
12 0re 9648 . . . . . . . . . . . . 13  |-  0  e.  RR
13 ltlen 9740 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
1412, 3, 13sylancr 670 . . . . . . . . . . . 12  |-  ( B  e.  ( 0 [,] 1 )  ->  (
0  <  B  <->  ( 0  <_  B  /\  B  =/=  0 ) ) )
1514biimpar 488 . . . . . . . . . . 11  |-  ( ( B  e.  ( 0 [,] 1 )  /\  ( 0  <_  B  /\  B  =/=  0
) )  ->  0  <  B )
16153impb 1205 . . . . . . . . . 10  |-  ( ( B  e.  ( 0 [,] 1 )  /\  0  <_  B  /\  B  =/=  0 )  ->  0  <  B )
17163com23 1215 . . . . . . . . 9  |-  ( ( B  e.  ( 0 [,] 1 )  /\  B  =/=  0  /\  0  <_  B )  ->  0  <  B )
1811, 17mpd3an3 1367 . . . . . . . 8  |-  ( ( B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <  B )
19183adant1 1027 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <  B )
20 divge0 10481 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  0  <_  ( A  /  B ) )
212, 9, 4, 19, 20syl22anc 1270 . . . . . 6  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <_  ( A  /  B ) )
2221adantr 467 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0
)  /\  A  <_  B )  ->  0  <_  ( A  /  B ) )
23 1red 9663 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
1  e.  RR )
24 ledivmul 10488 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( A  /  B )  <_  1  <->  A  <_  ( B  x.  1 ) ) )
252, 23, 4, 19, 24syl112anc 1273 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( ( A  /  B )  <_  1  <->  A  <_  ( B  x.  1 ) ) )
26 ax-1rid 9614 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
2726breq2d 4417 . . . . . . . 8  |-  ( B  e.  RR  ->  ( A  <_  ( B  x.  1 )  <->  A  <_  B ) )
284, 27syl 17 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  <_  ( B  x.  1 )  <-> 
A  <_  B )
)
2925, 28bitr2d 258 . . . . . 6  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  <_  B  <->  ( A  /  B )  <_  1 ) )
3029biimpa 487 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0
)  /\  A  <_  B )  ->  ( A  /  B )  <_  1
)
317, 22, 303jca 1189 . . . 4  |-  ( ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0
)  /\  A  <_  B )  ->  ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
)  /\  ( A  /  B )  <_  1
) )
3231ex 436 . . 3  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  <_  B  ->  ( ( A  /  B )  e.  RR  /\  0  <_  ( A  /  B )  /\  ( A  /  B )  <_ 
1 ) ) )
33 simp3 1011 . . . 4  |-  ( ( ( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B )  /\  ( A  /  B )  <_ 
1 )  ->  ( A  /  B )  <_ 
1 )
3433, 29syl5ibr 225 . . 3  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
)  /\  ( A  /  B )  <_  1
)  ->  A  <_  B ) )
3532, 34impbid 194 . 2  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  <_  B  <->  ( ( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B )  /\  ( A  /  B )  <_ 
1 ) ) )
36 1re 9647 . . 3  |-  1  e.  RR
3712, 36elicc2i 11707 . 2  |-  ( ( A  /  B )  e.  ( 0 [,] 1 )  <->  ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
)  /\  ( A  /  B )  <_  1
) )
3835, 37syl6bbr 267 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 188    /\ wa 371    /\ w3a 986    e. wcel 1889    =/= wne 2624   class class class wbr 4405  (class class class)co 6295   RRcr 9543   0cc0 9544   1c1 9545    x. cmul 9549    < clt 9680    <_ cle 9681    / cdiv 10276   [,]cicc 11645
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-cnex 9600  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  df-icc 11649
This theorem is referenced by:  cndprob01  29280
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