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Theorem icccntri 9135
Description: Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
icccntri.1  |-  A  e.  RR
icccntri.2  |-  B  e.  RR
icccntri.3  |-  R  e.  RR+
icccntri.4  |-  ( A  /  R )  =  C
icccntri.5  |-  ( B  /  R )  =  D
Assertion
Ref Expression
icccntri  |-  ( X  e.  ( A [,] B )  ->  ( X  /  R )  e.  ( C [,] D
) )

Proof of Theorem icccntri
StepHypRef Expression
1 icccntri.1 . . . 4  |-  A  e.  RR
2 icccntri.2 . . . 4  |-  B  e.  RR
3 iccssre 9090 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3mp2an 417 . . 3  |-  ( A [,] B )  C_  RR
54sseli 3005 . 2  |-  ( X  e.  ( A [,] B )  ->  X  e.  RR )
6 icccntri.3 . . . 4  |-  R  e.  RR+
7 icccntri.4 . . . . . 6  |-  ( A  /  R )  =  C
8 icccntri.5 . . . . . 6  |-  ( B  /  R )  =  D
97, 8icccntr 9134 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  /  R )  e.  ( C [,] D ) ) )
101, 2, 9mpanl12 427 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  e.  ( A [,] B )  <-> 
( X  /  R
)  e.  ( C [,] D ) ) )
116, 10mpan2 416 . . 3  |-  ( X  e.  RR  ->  ( X  e.  ( A [,] B )  <->  ( X  /  R )  e.  ( C [,] D ) ) )
1211biimpd 142 . 2  |-  ( X  e.  RR  ->  ( X  e.  ( A [,] B )  ->  ( X  /  R )  e.  ( C [,] D
) ) )
135, 12mpcom 36 1  |-  ( X  e.  ( A [,] B )  ->  ( X  /  R )  e.  ( C [,] D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434    C_ wss 2983  (class class class)co 5564   RRcr 7078    / cdiv 7863   RR+crp 8851   [,]cicc 9026
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 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-mulrcl 7173  ax-addcom 7174  ax-mulcom 7175  ax-addass 7176  ax-mulass 7177  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-1rid 7181  ax-0id 7182  ax-rnegex 7183  ax-precex 7184  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-apti 7189  ax-pre-ltadd 7190  ax-pre-mulgt0 7191  ax-pre-mulext 7192
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 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-id 4077  df-po 4080  df-iso 4081  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-reap 7778  df-ap 7785  df-div 7864  df-rp 8852  df-icc 9030
This theorem is referenced by: (None)
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