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Theorem iccshftl 10996
Description: Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
iccshftl.1  |-  ( A  -  R )  =  C
iccshftl.2  |-  ( B  -  R )  =  D
Assertion
Ref Expression
iccshftl  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  ( A [,] B )  <-> 
( X  -  R
)  e.  ( C [,] D ) ) )

Proof of Theorem iccshftl
StepHypRef Expression
1 simpl 444 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  X  e.  RR )
2 resubcl 9329 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  -  R
)  e.  RR )
31, 22thd 232 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  e.  RR  <->  ( X  -  R )  e.  RR ) )
43adantl 453 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  RR  <->  ( X  -  R )  e.  RR ) )
5 lesub1 9486 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR )  ->  ( A  <_  X  <->  ( A  -  R )  <_  ( X  -  R )
) )
653expb 1154 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( A  <_  X  <->  ( A  -  R )  <_  ( X  -  R )
) )
76adantlr 696 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( A  <_  X  <->  ( A  -  R )  <_  ( X  -  R ) ) )
8 iccshftl.1 . . . . 5  |-  ( A  -  R )  =  C
98breq1i 4187 . . . 4  |-  ( ( A  -  R )  <_  ( X  -  R )  <->  C  <_  ( X  -  R ) )
107, 9syl6bb 253 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( A  <_  X  <->  C  <_  ( X  -  R ) ) )
11 lesub1 9486 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR )  ->  ( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R )
) )
12113expb 1154 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R )
) )
1312an12s 777 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R )
) )
1413adantll 695 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R ) ) )
15 iccshftl.2 . . . . 5  |-  ( B  -  R )  =  D
1615breq2i 4188 . . . 4  |-  ( ( X  -  R )  <_  ( B  -  R )  <->  ( X  -  R )  <_  D
)
1714, 16syl6bb 253 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  <_  B  <->  ( X  -  R )  <_  D ) )
184, 10, 173anbi123d 1254 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( ( X  e.  RR  /\  A  <_  X  /\  X  <_  B
)  <->  ( ( X  -  R )  e.  RR  /\  C  <_ 
( X  -  R
)  /\  ( X  -  R )  <_  D
) ) )
19 elicc2 10939 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2019adantr 452 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
21 resubcl 9329 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  -  R
)  e.  RR )
228, 21syl5eqelr 2497 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
23 resubcl 9329 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  -  R
)  e.  RR )
2415, 23syl5eqelr 2497 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
25 elicc2 10939 . . . . 5  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( ( X  -  R )  e.  ( C [,] D )  <-> 
( ( X  -  R )  e.  RR  /\  C  <_  ( X  -  R )  /\  ( X  -  R )  <_  D ) ) )
2622, 24, 25syl2an 464 . . . 4  |-  ( ( ( A  e.  RR  /\  R  e.  RR )  /\  ( B  e.  RR  /\  R  e.  RR ) )  -> 
( ( X  -  R )  e.  ( C [,] D )  <-> 
( ( X  -  R )  e.  RR  /\  C  <_  ( X  -  R )  /\  ( X  -  R )  <_  D ) ) )
2726anandirs 805 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  R  e.  RR )  ->  ( ( X  -  R )  e.  ( C [,] D
)  <->  ( ( X  -  R )  e.  RR  /\  C  <_ 
( X  -  R
)  /\  ( X  -  R )  <_  D
) ) )
2827adantrl 697 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( ( X  -  R )  e.  ( C [,] D )  <-> 
( ( X  -  R )  e.  RR  /\  C  <_  ( X  -  R )  /\  ( X  -  R )  <_  D ) ) )
2918, 20, 283bitr4d 277 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  ( A [,] B )  <-> 
( X  -  R
)  e.  ( C [,] D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4180  (class class class)co 6048   RRcr 8953    <_ cle 9085    - cmin 9255   [,]cicc 10883
This theorem is referenced by:  iccshftli  10997  iccf1o  11003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-icc 10887
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