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Theorem iccshftri 10014
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 9974 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3mp2an 426 . . 3  |-  ( A [,] B )  C_  RR
54sseli 3166 . 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 10013 . . . . 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 436 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  +  R
)  e.  ( C [,] D ) ) )
116, 10mpan2 425 . . 3  |-  ( X  e.  RR  ->  ( X  e.  ( A [,] B )  <->  ( X  +  R )  e.  ( C [,] D ) ) )
1211biimpd 144 . 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 104    <-> wb 105    = wceq 1364    e. wcel 2160    C_ wss 3144  (class class class)co 5891   RRcr 7829    + caddc 7833   [,]cicc 9910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-i2m1 7935  ax-0id 7938  ax-rnegex 7939  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-ltadd 7946
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-icc 9914
This theorem is referenced by: (None)
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