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Theorem iccdil 11428
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 457 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  ->  X  e.  RR )
2 rpre 11002 . . . . . 6  |-  ( R  e.  RR+  ->  R  e.  RR )
3 remulcl 9372 . . . . . 6  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  x.  R
)  e.  RR )
42, 3sylan2 474 . . . . 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 466 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  RR  <->  ( X  x.  R )  e.  RR ) )
7 elrp 10998 . . . . . . 7  |-  ( R  e.  RR+  <->  ( R  e.  RR  /\  0  < 
R ) )
8 lemul1 10186 . . . . . . 7  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R ) ) )
97, 8syl3an3b 1256 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR+ )  ->  ( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R )
) )
1093expb 1188 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R )
) )
1110adantlr 714 . . . 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 4304 . . . 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 10186 . . . . . . . 8  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R ) ) )
167, 15syl3an3b 1256 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR+ )  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
17163expb 1188 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
1817an12s 799 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
1918adantll 713 . . . 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 4305 . . . 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 1289 . 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 11365 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2524adantr 465 . 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 9372 . . . . . . 7  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  x.  R
)  e.  RR )
2712, 26syl5eqelr 2528 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
28 remulcl 9372 . . . . . . 7  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  x.  R
)  e.  RR )
2920, 28syl5eqelr 2528 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
30 elicc2 11365 . . . . . 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 477 . . . . 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 827 . . . 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 474 . . 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 715 . 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 965    = wceq 1369    e. wcel 1756   class class class wbr 4297  (class class class)co 6096   RRcr 9286   0cc0 9287    x. cmul 9292    < clt 9423    <_ cle 9424   RR+crp 10996   [,]cicc 11308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-rp 10997  df-icc 11312
This theorem is referenced by:  iccdili  11429  lincmb01cmp  11433  iccf1o  11434
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