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Theorem icccntr 9776
Description: Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
icccntr.1  |-  ( A  /  R )  =  C
icccntr.2  |-  ( B  /  R )  =  D
Assertion
Ref Expression
icccntr  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  /  R )  e.  ( C [,] D ) ) )

Proof of Theorem icccntr
StepHypRef Expression
1 simpl 108 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  ->  X  e.  RR )
2 rerpdivcl 9465 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  /  R
)  e.  RR )
31, 22thd 174 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  e.  RR  <->  ( X  /  R )  e.  RR ) )
43adantl 275 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  RR  <->  ( X  /  R )  e.  RR ) )
5 elrp 9436 . . . . . . 7  |-  ( R  e.  RR+  <->  ( R  e.  RR  /\  0  < 
R ) )
6 lediv1 8620 . . . . . . 7  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
75, 6syl3an3b 1254 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR+ )  ->  ( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
873expb 1182 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
98adantlr 468 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
10 icccntr.1 . . . . 5  |-  ( A  /  R )  =  C
1110breq1i 3931 . . . 4  |-  ( ( A  /  R )  <_  ( X  /  R )  <->  C  <_  ( X  /  R ) )
129, 11syl6bb 195 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( A  <_  X  <->  C  <_  ( X  /  R ) ) )
13 lediv1 8620 . . . . . . . 8  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
145, 13syl3an3b 1254 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR+ )  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
15143expb 1182 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
1615an12s 554 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
1716adantll 467 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
18 icccntr.2 . . . . 5  |-  ( B  /  R )  =  D
1918breq2i 3932 . . . 4  |-  ( ( X  /  R )  <_  ( B  /  R )  <->  ( X  /  R )  <_  D
)
2017, 19syl6bb 195 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  <_  B  <->  ( X  /  R )  <_  D
) )
214, 12, 203anbi123d 1290 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  (
( X  e.  RR  /\  A  <_  X  /\  X  <_  B )  <->  ( ( X  /  R )  e.  RR  /\  C  <_ 
( X  /  R
)  /\  ( X  /  R )  <_  D
) ) )
22 elicc2 9714 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2322adantr 274 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  e.  RR  /\  A  <_  X  /\  X  <_  B
) ) )
24 rerpdivcl 9465 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR+ )  -> 
( A  /  R
)  e.  RR )
2510, 24eqeltrrid 2225 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR+ )  ->  C  e.  RR )
26 rerpdivcl 9465 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR+ )  -> 
( B  /  R
)  e.  RR )
2718, 26eqeltrrid 2225 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR+ )  ->  D  e.  RR )
28 elicc2 9714 . . . . 5  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( ( X  /  R )  e.  ( C [,] D )  <-> 
( ( X  /  R )  e.  RR  /\  C  <_  ( X  /  R )  /\  ( X  /  R )  <_  D ) ) )
2925, 27, 28syl2an 287 . . . 4  |-  ( ( ( A  e.  RR  /\  R  e.  RR+ )  /\  ( B  e.  RR  /\  R  e.  RR+ )
)  ->  ( ( X  /  R )  e.  ( C [,] D
)  <->  ( ( X  /  R )  e.  RR  /\  C  <_ 
( X  /  R
)  /\  ( X  /  R )  <_  D
) ) )
3029anandirs 582 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  R  e.  RR+ )  ->  ( ( X  /  R )  e.  ( C [,] D
)  <->  ( ( X  /  R )  e.  RR  /\  C  <_ 
( X  /  R
)  /\  ( X  /  R )  <_  D
) ) )
3130adantrl 469 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  (
( X  /  R
)  e.  ( C [,] D )  <->  ( ( X  /  R )  e.  RR  /\  C  <_ 
( X  /  R
)  /\  ( X  /  R )  <_  D
) ) )
3221, 23, 313bitr4d 219 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  /  R )  e.  ( C [,] D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3924  (class class class)co 5767   RRcr 7612   0cc0 7613    < clt 7793    <_ cle 7794    / cdiv 8425   RR+crp 9434   [,]cicc 9667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-rp 9435  df-icc 9671
This theorem is referenced by:  icccntri  9777
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