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Theorem iccshftri 9900
Description: Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
iccshftri.1  |-  A  e.  RR
iccshftri.2  |-  B  e.  RR
iccshftri.3  |-  R  e.  RR
iccshftri.4  |-  ( A  +  R )  =  C
iccshftri.5  |-  ( B  +  R )  =  D
Assertion
Ref Expression
iccshftri  |-  ( X  e.  ( A [,] B )  ->  ( X  +  R )  e.  ( C [,] D
) )

Proof of Theorem iccshftri
StepHypRef Expression
1 iccshftri.1 . . . 4  |-  A  e.  RR
2 iccshftri.2 . . . 4  |-  B  e.  RR
3 iccssre 9860 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3mp2an 423 . . 3  |-  ( A [,] B )  C_  RR
54sseli 3124 . 2  |-  ( X  e.  ( A [,] B )  ->  X  e.  RR )
6 iccshftri.3 . . . 4  |-  R  e.  RR
7 iccshftri.4 . . . . . 6  |-  ( A  +  R )  =  C
8 iccshftri.5 . . . . . 6  |-  ( B  +  R )  =  D
97, 8iccshftr 9899 . . . . 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 1335    e. wcel 2128    C_ wss 3102  (class class class)co 5825   RRcr 7732    + caddc 7736   [,]cicc 9796
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-addcom 7833  ax-addass 7835  ax-i2m1 7838  ax-0id 7841  ax-rnegex 7842  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-ltadd 7849
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-id 4254  df-po 4257  df-iso 4258  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-iota 5136  df-fun 5173  df-fv 5179  df-ov 5828  df-oprab 5829  df-mpo 5830  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-icc 9800
This theorem is referenced by: (None)
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