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Theorem icccntr 9415
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 107 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  ->  X  e.  RR )
2 rerpdivcl 9162 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  /  R
)  e.  RR )
31, 22thd 173 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  e.  RR  <->  ( X  /  R )  e.  RR ) )
43adantl 271 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  RR  <->  ( X  /  R )  e.  RR ) )
5 elrp 9134 . . . . . . 7  |-  ( R  e.  RR+  <->  ( R  e.  RR  /\  0  < 
R ) )
6 lediv1 8328 . . . . . . 7  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
75, 6syl3an3b 1212 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR+ )  ->  ( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
873expb 1144 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
98adantlr 461 . . . 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 3852 . . . 4  |-  ( ( A  /  R )  <_  ( X  /  R )  <->  C  <_  ( X  /  R ) )
129, 11syl6bb 194 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( A  <_  X  <->  C  <_  ( X  /  R ) ) )
13 lediv1 8328 . . . . . . . 8  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
145, 13syl3an3b 1212 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR+ )  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
15143expb 1144 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
1615an12s 532 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
1716adantll 460 . . . 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 3853 . . . 4  |-  ( ( X  /  R )  <_  ( B  /  R )  <->  ( X  /  R )  <_  D
)
2017, 19syl6bb 194 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  <_  B  <->  ( X  /  R )  <_  D
) )
214, 12, 203anbi123d 1248 . 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 9354 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2322adantr 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
) ) )
24 rerpdivcl 9162 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR+ )  -> 
( A  /  R
)  e.  RR )
2510, 24syl5eqelr 2175 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR+ )  ->  C  e.  RR )
26 rerpdivcl 9162 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR+ )  -> 
( B  /  R
)  e.  RR )
2718, 26syl5eqelr 2175 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR+ )  ->  D  e.  RR )
28 elicc2 9354 . . . . 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 283 . . . 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 560 . . 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 462 . 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 218 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   RRcr 7347   0cc0 7348    < clt 7520    <_ cle 7521    / cdiv 8137   RR+crp 9132   [,]cicc 9307
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-po 4123  df-iso 4124  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-rp 9133  df-icc 9311
This theorem is referenced by:  icccntri  9416
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