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Theorem icccntr 10791
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 443 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  ->  X  e.  RR )
2 rerpdivcl 10397 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  /  R
)  e.  RR )
31, 22thd 231 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  e.  RR  <->  ( X  /  R )  e.  RR ) )
43adantl 452 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  RR  <->  ( X  /  R )  e.  RR ) )
5 elrp 10372 . . . . . . 7  |-  ( R  e.  RR+  <->  ( R  e.  RR  /\  0  < 
R ) )
6 lediv1 9637 . . . . . . 7  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
75, 6syl3an3b 1220 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR+ )  ->  ( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
873expb 1152 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( A  <_  X  <->  ( A  /  R )  <_  ( X  /  R ) ) )
98adantlr 695 . . . 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 4046 . . . 4  |-  ( ( A  /  R )  <_  ( X  /  R )  <->  C  <_  ( X  /  R ) )
129, 11syl6bb 252 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( A  <_  X  <->  C  <_  ( X  /  R ) ) )
13 lediv1 9637 . . . . . . . 8  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  ( R  e.  RR  /\  0  <  R ) )  -> 
( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
145, 13syl3an3b 1220 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR+ )  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
15143expb 1152 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
1615an12s 776 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR+ )
)  ->  ( X  <_  B  <->  ( X  /  R )  <_  ( B  /  R ) ) )
1716adantll 694 . . . 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 4047 . . . 4  |-  ( ( X  /  R )  <_  ( B  /  R )  <->  ( X  /  R )  <_  D
)
2017, 19syl6bb 252 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  <_  B  <->  ( X  /  R )  <_  D
) )
214, 12, 203anbi123d 1252 . 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 10731 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2322adantr 451 . 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 10397 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR+ )  -> 
( A  /  R
)  e.  RR )
2510, 24syl5eqelr 2381 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR+ )  ->  C  e.  RR )
26 rerpdivcl 10397 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR+ )  -> 
( B  /  R
)  e.  RR )
2718, 26syl5eqelr 2381 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR+ )  ->  D  e.  RR )
28 elicc2 10731 . . . . 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 463 . . . 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 804 . . 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 696 . 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 276 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   RRcr 8752   0cc0 8753    < clt 8883    <_ cle 8884    / cdiv 9439   RR+crp 10370   [,]cicc 10675
This theorem is referenced by:  icccntri  10792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-rp 10371  df-icc 10679
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