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Theorem unitdivcld 28044
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 28040 . . . . . . . 8  |-  ( A  e.  ( 0 [,] 1 )  ->  A  e.  RR )
213ad2ant1 1017 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  ->  A  e.  RR )
3 elunitrn 28040 . . . . . . . 8  |-  ( B  e.  ( 0 [,] 1 )  ->  B  e.  RR )
433ad2ant2 1018 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  ->  B  e.  RR )
5 simp3 998 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  ->  B  =/=  0 )
62, 4, 5redivcld 10393 . . . . . 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 28042 . . . . . . . 8  |-  ( A  e.  ( 0 [,] 1 )  ->  0  <_  A )
983ad2ant1 1017 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <_  A )
10 elunitge0 28042 . . . . . . . . . 10  |-  ( B  e.  ( 0 [,] 1 )  ->  0  <_  B )
1110adantr 465 . . . . . . . . 9  |-  ( ( B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <_  B )
12 0re 9613 . . . . . . . . . . . . 13  |-  0  e.  RR
13 ltlen 9703 . . . . . . . . . . . . 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 1192 . . . . . . . . . 10  |-  ( ( B  e.  ( 0 [,] 1 )  /\  0  <_  B  /\  B  =/=  0 )  ->  0  <  B )
17163com23 1202 . . . . . . . . 9  |-  ( ( B  e.  ( 0 [,] 1 )  /\  B  =/=  0  /\  0  <_  B )  ->  0  <  B )
1811, 17mpd3an3 1325 . . . . . . . 8  |-  ( ( B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <  B )
19183adant1 1014 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
0  <  B )
20 divge0 10432 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  0  <_  ( A  /  B ) )
212, 9, 4, 19, 20syl22anc 1229 . . . . . 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 1red 9628 . . . . . . . 8  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
1  e.  RR )
24 ledivmul 10439 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( A  /  B )  <_  1  <->  A  <_  ( B  x.  1 ) ) )
252, 23, 4, 19, 24syl112anc 1232 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( ( A  /  B )  <_  1  <->  A  <_  ( B  x.  1 ) ) )
26 ax-1rid 9579 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
2726breq2d 4468 . . . . . . . 8  |-  ( B  e.  RR  ->  ( A  <_  ( B  x.  1 )  <->  A  <_  B ) )
284, 27syl 16 . . . . . . 7  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  <_  ( B  x.  1 )  <-> 
A  <_  B )
)
2925, 28bitr2d 254 . . . . . 6  |-  ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  -> 
( A  <_  B  <->  ( A  /  B )  <_  1 ) )
3029biimpa 484 . . . . 5  |-  ( ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0
)  /\  A  <_  B )  ->  ( A  /  B )  <_  1
)
317, 22, 303jca 1176 . . . 4  |-  ( ( ( A  e.  ( 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0
)  /\  A  <_  B )  ->  ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
)  /\  ( A  /  B )  <_  1
) )
3231ex 434 . . 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 998 . . . 4  |-  ( ( ( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B )  /\  ( A  /  B )  <_ 
1 )  ->  ( A  /  B )  <_ 
1 )
3433, 29syl5ibr 221 . . 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 191 . 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 9612 . . 3  |-  1  e.  RR
3712, 36elicc2i 11615 . 2  |-  ( ( A  /  B )  e.  ( 0 [,] 1 )  <->  ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
)  /\  ( A  /  B )  <_  1
) )
3835, 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 973    e. wcel 1819    =/= wne 2652   class class class wbr 4456  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514    < clt 9645    <_ cle 9646    / cdiv 10227   [,]cicc 11557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-icc 11561
This theorem is referenced by:  cndprob01  28571
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