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Theorem icccntri 9813
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 9767 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3mp2an 423 . . 3  |-  ( A [,] B )  C_  RR
54sseli 3097 . 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 9812 . . . . 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 433 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR+ )  -> 
( X  e.  ( A [,] B )  <-> 
( X  /  R
)  e.  ( C [,] D ) ) )
116, 10mpan2 422 . . 3  |-  ( X  e.  RR  ->  ( X  e.  ( A [,] B )  <->  ( X  /  R )  e.  ( C [,] D ) ) )
1211biimpd 143 . 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 103    <-> wb 104    = wceq 1332    e. wcel 1481    C_ wss 3075  (class class class)co 5781   RRcr 7642    / cdiv 8455   RR+crp 9469   [,]cicc 9703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-po 4225  df-iso 4226  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-rp 9470  df-icc 9707
This theorem is referenced by: (None)
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