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Theorem iccshftri 10770
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 10731 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3mp2an 653 . . 3  |-  ( A [,] B )  C_  RR
54sseli 3176 . 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 10769 . . . . 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 663 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  +  R
)  e.  ( C [,] D ) ) )
116, 10mpan2 652 . . 3  |-  ( X  e.  RR  ->  ( X  e.  ( A [,] B )  <->  ( X  +  R )  e.  ( C [,] D ) ) )
1211biimpd 198 . 2  |-  ( X  e.  RR  ->  ( X  e.  ( A [,] B )  ->  ( X  +  R )  e.  ( C [,] D
) ) )
135, 12mpcom 32 1  |-  ( X  e.  ( A [,] B )  ->  ( X  +  R )  e.  ( C [,] D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152  (class class class)co 5858   RRcr 8736    + caddc 8740   [,]cicc 10659
This theorem is referenced by:  pcoass  18522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-icc 10663
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