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Theorem iccdil 9148
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 107 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  ->  X  e.  RR )
2 rpre 8873 . . . . . 6  |-  ( R  e.  RR+  ->  R  e.  RR )
3 remulcl 7215 . . . . . 6  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  x.  R
)  e.  RR )
42, 3sylan2 280 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  x.  R
)  e.  RR )
51, 42thd 173 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  e.  RR  <->  ( X  x.  R )  e.  RR ) )
65adantl 271 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  RR  <->  ( X  x.  R )  e.  RR ) )
7 elrp 8869 . . . . . . 7  |-  ( R  e.  RR+  <->  ( R  e.  RR  /\  0  < 
R ) )
8 lemul1 7812 . . . . . . 7  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R ) ) )
97, 8syl3an3b 1208 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR+ )  ->  ( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R )
) )
1093expb 1140 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R )
) )
1110adantlr 461 . . . 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 3812 . . . 4  |-  ( ( A  x.  R )  <_  ( X  x.  R )  <->  C  <_  ( X  x.  R ) )
1411, 13syl6bb 194 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( A  <_  X  <->  C  <_  ( X  x.  R ) ) )
15 lemul1 7812 . . . . . . . 8  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R ) ) )
167, 15syl3an3b 1208 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR+ )  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
17163expb 1140 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
1817an12s 530 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
1918adantll 460 . . . 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 3813 . . . 4  |-  ( ( X  x.  R )  <_  ( B  x.  R )  <->  ( X  x.  R )  <_  D
)
2219, 21syl6bb 194 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  <_  B  <->  ( X  x.  R )  <_  D
) )
236, 14, 223anbi123d 1244 . 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 9089 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2524adantr 270 . 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 7215 . . . . . . 7  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  x.  R
)  e.  RR )
2712, 26syl5eqelr 2170 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
28 remulcl 7215 . . . . . . 7  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  x.  R
)  e.  RR )
2920, 28syl5eqelr 2170 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
30 elicc2 9089 . . . . . 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 283 . . . . 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 558 . . . 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 280 . . 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 462 . 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 218 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    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3805  (class class class)co 5563   RRcr 7094   0cc0 7095    x. cmul 7100    < clt 7267    <_ cle 7268   RR+crp 8867   [,]cicc 9042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206  ax-pre-mulgt0 7207
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-po 4079  df-iso 4080  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-rp 8868  df-icc 9046
This theorem is referenced by:  iccdili  9149  lincmb01cmp  9153  iccf1o  9154
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