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Theorem iccdil 11036
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 445 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  ->  X  e.  RR )
2 rpre 10620 . . . . . 6  |-  ( R  e.  RR+  ->  R  e.  RR )
3 remulcl 9077 . . . . . 6  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  x.  R
)  e.  RR )
42, 3sylan2 462 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  x.  R
)  e.  RR )
51, 42thd 233 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  e.  RR  <->  ( X  x.  R )  e.  RR ) )
65adantl 454 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  RR  <->  ( X  x.  R )  e.  RR ) )
7 elrp 10616 . . . . . . 7  |-  ( R  e.  RR+  <->  ( R  e.  RR  /\  0  < 
R ) )
8 lemul1 9864 . . . . . . 7  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R ) ) )
97, 8syl3an3b 1223 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR+ )  ->  ( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R )
) )
1093expb 1155 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( A  <_  X  <->  ( A  x.  R )  <_  ( X  x.  R )
) )
1110adantlr 697 . . . 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 4221 . . . 4  |-  ( ( A  x.  R )  <_  ( X  x.  R )  <->  C  <_  ( X  x.  R ) )
1411, 13syl6bb 254 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( A  <_  X  <->  C  <_  ( X  x.  R ) ) )
15 lemul1 9864 . . . . . . . 8  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R ) ) )
167, 15syl3an3b 1223 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR+ )  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
17163expb 1155 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
1817an12s 778 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  x.  R )  <_  ( B  x.  R )
) )
1918adantll 696 . . . 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 4222 . . . 4  |-  ( ( X  x.  R )  <_  ( B  x.  R )  <->  ( X  x.  R )  <_  D
)
2219, 21syl6bb 254 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  <_  B  <->  ( X  x.  R )  <_  D
) )
236, 14, 223anbi123d 1255 . 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 10977 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2524adantr 453 . 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 9077 . . . . . . 7  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  x.  R
)  e.  RR )
2712, 26syl5eqelr 2523 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
28 remulcl 9077 . . . . . . 7  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  x.  R
)  e.  RR )
2920, 28syl5eqelr 2523 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
30 elicc2 10977 . . . . . 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 465 . . . . 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 806 . . . 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 462 . . 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 698 . 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 278 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214  (class class class)co 6083   RRcr 8991   0cc0 8992    x. cmul 8997    < clt 9122    <_ cle 9123   RR+crp 10614   [,]cicc 10921
This theorem is referenced by:  iccdili  11037  lincmb01cmp  11040  iccf1o  11041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-rp 10615  df-icc 10925
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