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Theorem iccshftli 9787
Description: Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
iccshftli.1  |-  A  e.  RR
iccshftli.2  |-  B  e.  RR
iccshftli.3  |-  R  e.  RR
iccshftli.4  |-  ( A  -  R )  =  C
iccshftli.5  |-  ( B  -  R )  =  D
Assertion
Ref Expression
iccshftli  |-  ( X  e.  ( A [,] B )  ->  ( X  -  R )  e.  ( C [,] D
) )

Proof of Theorem iccshftli
StepHypRef Expression
1 iccshftli.1 . . . 4  |-  A  e.  RR
2 iccshftli.2 . . . 4  |-  B  e.  RR
3 iccssre 9745 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3mp2an 422 . . 3  |-  ( A [,] B )  C_  RR
54sseli 3093 . 2  |-  ( X  e.  ( A [,] B )  ->  X  e.  RR )
6 iccshftli.3 . . . 4  |-  R  e.  RR
7 iccshftli.4 . . . . . 6  |-  ( A  -  R )  =  C
8 iccshftli.5 . . . . . 6  |-  ( B  -  R )  =  D
97, 8iccshftl 9786 . . . . 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 432 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  -  R
)  e.  ( C [,] D ) ) )
116, 10mpan2 421 . . 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 1331    e. wcel 1480    C_ wss 3071  (class class class)co 5774   RRcr 7626    - cmin 7940   [,]cicc 9681
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-icc 9685
This theorem is referenced by: (None)
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