MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iccdili Unicode version

Theorem iccdili 10991
Description: Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
iccdili.1  |-  A  e.  RR
iccdili.2  |-  B  e.  RR
iccdili.3  |-  R  e.  RR+
iccdili.4  |-  ( A  x.  R )  =  C
iccdili.5  |-  ( B  x.  R )  =  D
Assertion
Ref Expression
iccdili  |-  ( X  e.  ( A [,] B )  ->  ( X  x.  R )  e.  ( C [,] D
) )

Proof of Theorem iccdili
StepHypRef Expression
1 iccdili.1 . . . 4  |-  A  e.  RR
2 iccdili.2 . . . 4  |-  B  e.  RR
3 iccssre 10948 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3mp2an 654 . . 3  |-  ( A [,] B )  C_  RR
54sseli 3304 . 2  |-  ( X  e.  ( A [,] B )  ->  X  e.  RR )
6 iccdili.3 . . . 4  |-  R  e.  RR+
7 iccdili.4 . . . . . 6  |-  ( A  x.  R )  =  C
8 iccdili.5 . . . . . 6  |-  ( B  x.  R )  =  D
97, 8iccdil 10990 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR+ ) )  ->  ( X  e.  ( A [,] B )  <->  ( X  x.  R )  e.  ( C [,] D ) ) )
101, 2, 9mpanl12 664 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  e.  ( A [,] B )  <-> 
( X  x.  R
)  e.  ( C [,] D ) ) )
116, 10mpan2 653 . . 3  |-  ( X  e.  RR  ->  ( X  e.  ( A [,] B )  <->  ( X  x.  R )  e.  ( C [,] D ) ) )
1211biimpd 199 . 2  |-  ( X  e.  RR  ->  ( X  e.  ( A [,] B )  ->  ( X  x.  R )  e.  ( C [,] D
) ) )
135, 12mpcom 34 1  |-  ( X  e.  ( A [,] B )  ->  ( X  x.  R )  e.  ( C [,] D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3280  (class class class)co 6040   RRcr 8945    x. cmul 8951   RR+crp 10568   [,]cicc 10875
This theorem is referenced by:  pcoass  19002  cxpsqrlem  20546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-rp 10569  df-icc 10879
  Copyright terms: Public domain W3C validator