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Theorem icccntr 9098
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 8845 . . . . 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 8817 . . . . . . 7  |-  ( R  e.  RR+  <->  ( R  e.  RR  /\  0  < 
R ) )
6 lediv1 8014 . . . . . . 7  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
75, 6syl3an3b 1208 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR+ )  ->  ( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
873expb 1140 . . . . 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 3800 . . . 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 8014 . . . . . . . 8  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
145, 13syl3an3b 1208 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR+ )  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
15143expb 1140 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
1615an12s 530 . . . . 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 3801 . . . 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 1244 . 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 9037 . . 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 8845 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR+ )  -> 
( A  /  R
)  e.  RR )
2510, 24syl5eqelr 2167 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR+ )  ->  C  e.  RR )
26 rerpdivcl 8845 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR+ )  -> 
( B  /  R
)  e.  RR )
2718, 26syl5eqelr 2167 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR+ )  ->  D  e.  RR )
28 elicc2 9037 . . . . 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 558 . . 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 920    = wceq 1285    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   RRcr 7042   0cc0 7043    < clt 7215    <_ cle 7216    / cdiv 7827   RR+crp 8815   [,]cicc 8990
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156
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 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-po 4059  df-iso 4060  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-rp 8816  df-icc 8994
This theorem is referenced by:  icccntri  9099
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