MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iccdil Unicode version

Theorem iccdil 10773
Description: Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
iccdil.1  |-  ( A  x.  R )  =  C
iccdil.2  |-  ( B  x.  R )  =  D
Assertion
Ref Expression
iccdil  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  x.  R )  e.  ( C [,] D ) ) )

Proof of Theorem iccdil
StepHypRef Expression
1 simpl 443 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  ->  X  e.  RR )
2 rpre 10360 . . . . . 6  |-  ( R  e.  RR+  ->  R  e.  RR )
3 remulcl 8822 . . . . . 6  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  x.  R
)  e.  RR )
42, 3sylan2 460 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  x.  R
)  e.  RR )
51, 42thd 231 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  e.  RR  <->  ( X  x.  R )  e.  RR ) )
65adantl 452 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  RR  <->  ( X  x.  R )  e.  RR ) )
7 elrp 10356 . . . . . . 7  |-  ( R  e.  RR+  <->  ( R  e.  RR  /\  0  < 
R ) )
8 lemul1 9608 . . . . . . 7  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R ) ) )
97, 8syl3an3b 1220 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR+ )  ->  ( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R )
) )
1093expb 1152 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R )
) )
1110adantlr 695 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R )
) )
12 iccdil.1 . . . . 5  |-  ( A  x.  R )  =  C
1312breq1i 4030 . . . 4  |-  ( ( A  x.  R )  <_  ( X  x.  R )  <->  C  <_  ( X  x.  R ) )
1411, 13syl6bb 252 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( A  <_  X  <->  C  <_  ( X  x.  R ) ) )
15 lemul1 9608 . . . . . . . 8  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R ) ) )
167, 15syl3an3b 1220 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR+ )  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
17163expb 1152 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
1817an12s 776 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
1918adantll 694 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
20 iccdil.2 . . . . 5  |-  ( B  x.  R )  =  D
2120breq2i 4031 . . . 4  |-  ( ( X  x.  R )  <_  ( B  x.  R )  <->  ( X  x.  R )  <_  D
)
2219, 21syl6bb 252 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  <_  B  <->  ( X  x.  R )  <_  D
) )
236, 14, 223anbi123d 1252 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  (
( X  e.  RR  /\  A  <_  X  /\  X  <_  B )  <->  ( ( X  x.  R )  e.  RR  /\  C  <_ 
( X  x.  R
)  /\  ( X  x.  R )  <_  D
) ) )
24 elicc2 10715 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2524adantr 451 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  e.  RR  /\  A  <_  X  /\  X  <_  B
) ) )
26 remulcl 8822 . . . . . . 7  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  x.  R
)  e.  RR )
2712, 26syl5eqelr 2368 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
28 remulcl 8822 . . . . . . 7  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  x.  R
)  e.  RR )
2920, 28syl5eqelr 2368 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
30 elicc2 10715 . . . . . 6  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( ( X  x.  R )  e.  ( C [,] D )  <-> 
( ( X  x.  R )  e.  RR  /\  C  <_  ( X  x.  R )  /\  ( X  x.  R )  <_  D ) ) )
3127, 29, 30syl2an 463 . . . . 5  |-  ( ( ( A  e.  RR  /\  R  e.  RR )  /\  ( B  e.  RR  /\  R  e.  RR ) )  -> 
( ( X  x.  R )  e.  ( C [,] D )  <-> 
( ( X  x.  R )  e.  RR  /\  C  <_  ( X  x.  R )  /\  ( X  x.  R )  <_  D ) ) )
3231anandirs 804 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  R  e.  RR )  ->  ( ( X  x.  R )  e.  ( C [,] D
)  <->  ( ( X  x.  R )  e.  RR  /\  C  <_ 
( X  x.  R
)  /\  ( X  x.  R )  <_  D
) ) )
332, 32sylan2 460 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  R  e.  RR+ )  ->  ( ( X  x.  R )  e.  ( C [,] D
)  <->  ( ( X  x.  R )  e.  RR  /\  C  <_ 
( X  x.  R
)  /\  ( X  x.  R )  <_  D
) ) )
3433adantrl 696 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  (
( X  x.  R
)  e.  ( C [,] D )  <->  ( ( X  x.  R )  e.  RR  /\  C  <_ 
( X  x.  R
)  /\  ( X  x.  R )  <_  D
) ) )
3523, 25, 343bitr4d 276 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  x.  R )  e.  ( C [,] D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   RRcr 8736   0cc0 8737    x. cmul 8742    < clt 8867    <_ cle 8868   RR+crp 10354   [,]cicc 10659
This theorem is referenced by:  iccdili  10774  lincmb01cmp  10777  iccf1o  10778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-rp 10355  df-icc 10663
  Copyright terms: Public domain W3C validator