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Theorem divelunit 12314
Description: A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
Assertion
Ref Expression
divelunit |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) e. ( 0 [,] 1 ) <-> A <_ B ) )
Proof of Theorem divelunit
StepHypRef Expression
1 0re 10040 . . . 4 |- 0 e. RR
2 1re 10039 . . . 4 |- 1 e. RR
31, 2elicc2i 12239 . . 3 |- ( ( A / B ) e. ( 0 [,] 1 ) <-> ( ( A / B ) e. RR /\ 0 <_ ( A / B ) /\ ( A / B ) <_ 1 ) )
4 df-3an 1039 . . 3 |- ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) /\ ( A / B ) <_ 1 ) <-> ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) /\ ( A / B ) <_ 1 ) )
53, 4bitri 264 . 2 |- ( ( A / B ) e. ( 0 [,] 1 ) <-> ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) /\ ( A / B ) <_ 1 ) )
6 ledivmul 10899 . . . . 5 |- ( ( A e. RR /\ 1 e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) <_ 1 <-> A <_ ( B x. 1 ) ) )
72, 6mp3an2 1412 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) <_ 1 <-> A <_ ( B x. 1 ) ) )
87adantlr 751 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) <_ 1 <-> A <_ ( B x. 1 ) ) )
9 simpll 790 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> A e. RR )
10 simprl 794 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> B e. RR )
11 gt0ne0 10493 . . . . . . 7 |- ( ( B e. RR /\ 0 < B ) -> B =/= 0 )
1211adantl 482 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> B =/= 0 )
139, 10, 12redivcld 10853 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A / B ) e. RR )
14 divge0 10892 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 <_ ( A / B ) )
1513, 14jca 554 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) )
1615biantrurd 529 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) <_ 1 <-> ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) /\ ( A / B ) <_ 1 ) ) )
17 recn 10026 . . . . . 6 |- ( B e. RR -> B e. CC )
1817ad2antrl 764 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> B e. CC )
1918mulid1d 10057 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( B x. 1 ) = B )
2019breq2d 4665 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ ( B x. 1 ) <-> A <_ B ) )
218, 16, 203bitr3d 298 . 2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) /\ ( A / B ) <_ 1 ) <-> A <_ B ) )
225, 21syl5bb 272 1 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) e. ( 0 [,] 1 ) <-> A <_ B ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   [,]cicc 12178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-icc 12182
This theorem is referenced by:  brbtwn2  25785  axsegconlem7  25803  axcontlem2  25845  axcontlem4  25847  axcontlem7  25850  axcontlem8  25851
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