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

Theorem iccdil 11419
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 454 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  ->  X  e.  RR )
2 rpre 10993 . . . . . 6  |-  ( R  e.  RR+  ->  R  e.  RR )
3 remulcl 9363 . . . . . 6  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  x.  R
)  e.  RR )
42, 3sylan2 471 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  x.  R
)  e.  RR )
51, 42thd 240 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  e.  RR  <->  ( X  x.  R )  e.  RR ) )
65adantl 463 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  RR  <->  ( X  x.  R )  e.  RR ) )
7 elrp 10989 . . . . . . 7  |-  ( R  e.  RR+  <->  ( R  e.  RR  /\  0  < 
R ) )
8 lemul1 10177 . . . . . . 7  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R ) ) )
97, 8syl3an3b 1251 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR+ )  ->  ( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R )
) )
1093expb 1183 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R )
) )
1110adantlr 709 . . . 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 4296 . . . 4  |-  ( ( A  x.  R )  <_  ( X  x.  R )  <->  C  <_  ( X  x.  R ) )
1411, 13syl6bb 261 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( A  <_  X  <->  C  <_  ( X  x.  R ) ) )
15 lemul1 10177 . . . . . . . 8  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R ) ) )
167, 15syl3an3b 1251 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR+ )  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
17163expb 1183 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
1817an12s 794 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
1918adantll 708 . . . 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 4297 . . . 4  |-  ( ( X  x.  R )  <_  ( B  x.  R )  <->  ( X  x.  R )  <_  D
)
2219, 21syl6bb 261 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  <_  B  <->  ( X  x.  R )  <_  D
) )
236, 14, 223anbi123d 1284 . 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 11356 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2524adantr 462 . 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 9363 . . . . . . 7  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  x.  R
)  e.  RR )
2712, 26syl5eqelr 2526 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
28 remulcl 9363 . . . . . . 7  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  x.  R
)  e.  RR )
2920, 28syl5eqelr 2526 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
30 elicc2 11356 . . . . . 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 474 . . . . 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 822 . . . 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 471 . . 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 710 . 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 285 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   class class class wbr 4289  (class class class)co 6090   RRcr 9277   0cc0 9278    x. cmul 9283    < clt 9414    <_ cle 9415   RR+crp 10987   [,]cicc 11299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-rp 10988  df-icc 11303
This theorem is referenced by:  iccdili  11420  lincmb01cmp  11424  iccf1o  11425
  Copyright terms: Public domain W3C validator