ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  icccntr Unicode version

Theorem icccntr 9905
Description: Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
icccntr.1  |-  ( A  /  R )  =  C
icccntr.2  |-  ( B  /  R )  =  D
Assertion
Ref Expression
icccntr  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  /  R )  e.  ( C [,] D ) ) )

Proof of Theorem icccntr
StepHypRef Expression
1 simpl 108 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  ->  X  e.  RR )
2 rerpdivcl 9592 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  /  R
)  e.  RR )
31, 22thd 174 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  e.  RR  <->  ( X  /  R )  e.  RR ) )
43adantl 275 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  RR  <->  ( X  /  R )  e.  RR ) )
5 elrp 9563 . . . . . . 7  |-  ( R  e.  RR+  <->  ( R  e.  RR  /\  0  < 
R ) )
6 lediv1 8741 . . . . . . 7  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
75, 6syl3an3b 1258 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR+ )  ->  ( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
873expb 1186 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
98adantlr 469 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
10 icccntr.1 . . . . 5  |-  ( A  /  R )  =  C
1110breq1i 3973 . . . 4  |-  ( ( A  /  R )  <_  ( X  /  R )  <->  C  <_  ( X  /  R ) )
129, 11bitrdi 195 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( A  <_  X  <->  C  <_  ( X  /  R ) ) )
13 lediv1 8741 . . . . . . . 8  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
145, 13syl3an3b 1258 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR+ )  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
15143expb 1186 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
1615an12s 555 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
1716adantll 468 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
18 icccntr.2 . . . . 5  |-  ( B  /  R )  =  D
1918breq2i 3974 . . . 4  |-  ( ( X  /  R )  <_  ( B  /  R )  <->  ( X  /  R )  <_  D
)
2017, 19bitrdi 195 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  <_  B  <->  ( X  /  R )  <_  D
) )
214, 12, 203anbi123d 1294 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  (
( X  e.  RR  /\  A  <_  X  /\  X  <_  B )  <->  ( ( X  /  R )  e.  RR  /\  C  <_ 
( X  /  R
)  /\  ( X  /  R )  <_  D
) ) )
22 elicc2 9843 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2322adantr 274 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  e.  RR  /\  A  <_  X  /\  X  <_  B
) ) )
24 rerpdivcl 9592 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR+ )  -> 
( A  /  R
)  e.  RR )
2510, 24eqeltrrid 2245 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR+ )  ->  C  e.  RR )
26 rerpdivcl 9592 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR+ )  -> 
( B  /  R
)  e.  RR )
2718, 26eqeltrrid 2245 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR+ )  ->  D  e.  RR )
28 elicc2 9843 . . . . 5  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( ( X  /  R )  e.  ( C [,] D )  <-> 
( ( X  /  R )  e.  RR  /\  C  <_  ( X  /  R )  /\  ( X  /  R )  <_  D ) ) )
2925, 27, 28syl2an 287 . . . 4  |-  ( ( ( A  e.  RR  /\  R  e.  RR+ )  /\  ( B  e.  RR  /\  R  e.  RR+ )
)  ->  ( ( X  /  R )  e.  ( C [,] D
)  <->  ( ( X  /  R )  e.  RR  /\  C  <_ 
( X  /  R
)  /\  ( X  /  R )  <_  D
) ) )
3029anandirs 583 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  R  e.  RR+ )  ->  ( ( X  /  R )  e.  ( C [,] D
)  <->  ( ( X  /  R )  e.  RR  /\  C  <_ 
( X  /  R
)  /\  ( X  /  R )  <_  D
) ) )
3130adantrl 470 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  (
( X  /  R
)  e.  ( C [,] D )  <->  ( ( X  /  R )  e.  RR  /\  C  <_ 
( X  /  R
)  /\  ( X  /  R )  <_  D
) ) )
3221, 23, 313bitr4d 219 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  /  R )  e.  ( C [,] D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128   class class class wbr 3966  (class class class)co 5825   RRcr 7732   0cc0 7733    < clt 7913    <_ cle 7914    / cdiv 8546   RR+crp 9561   [,]cicc 9796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-mulrcl 7832  ax-addcom 7833  ax-mulcom 7834  ax-addass 7835  ax-mulass 7836  ax-distr 7837  ax-i2m1 7838  ax-0lt1 7839  ax-1rid 7840  ax-0id 7841  ax-rnegex 7842  ax-precex 7843  ax-cnre 7844  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-apti 7848  ax-pre-ltadd 7849  ax-pre-mulgt0 7850  ax-pre-mulext 7851
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-id 4254  df-po 4257  df-iso 4258  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-iota 5136  df-fun 5173  df-fv 5179  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-sub 8049  df-neg 8050  df-reap 8451  df-ap 8458  df-div 8547  df-rp 9562  df-icc 9800
This theorem is referenced by:  icccntri  9906
  Copyright terms: Public domain W3C validator