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Theorem iccshftli 9748
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 9706 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3mp2an 422 . . 3  |-  ( A [,] B )  C_  RR
54sseli 3063 . 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 9747 . . . . 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 1316    e. wcel 1465    C_ wss 3041  (class class class)co 5742   RRcr 7587    - cmin 7901   [,]cicc 9642
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-icc 9646
This theorem is referenced by: (None)
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