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Theorem iccshftl 11627
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 455 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  X  e.  RR )
2 resubcl 9839 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  -  R
)  e.  RR )
31, 22thd 240 . . . 4  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  ( X  e.  RR  <->  ( X  -  R )  e.  RR ) )
43adantl 464 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  RR  <->  ( X  -  R )  e.  RR ) )
5 lesub1 10007 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR )  ->  ( A  <_  X  <->  ( A  -  R )  <_  ( X  -  R )
) )
653expb 1198 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( A  <_  X  <->  ( A  -  R )  <_  ( X  -  R )
) )
76adantlr 713 . . . 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 4401 . . . 4  |-  ( ( A  -  R )  <_  ( X  -  R )  <->  C  <_  ( X  -  R ) )
107, 9syl6bb 261 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( A  <_  X  <->  C  <_  ( X  -  R ) ) )
11 lesub1 10007 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR )  ->  ( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R )
) )
12113expb 1198 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R )
) )
1312an12s 802 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R )
) )
1413adantll 712 . . . 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 4402 . . . 4  |-  ( ( X  -  R )  <_  ( B  -  R )  <->  ( X  -  R )  <_  D
)
1714, 16syl6bb 261 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  <_  B  <->  ( X  -  R )  <_  D ) )
184, 10, 173anbi123d 1301 . 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 11560 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2019adantr 463 . 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 9839 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  -  R
)  e.  RR )
228, 21syl5eqelr 2495 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
23 resubcl 9839 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  -  R
)  e.  RR )
2415, 23syl5eqelr 2495 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
25 elicc2 11560 . . . . 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 475 . . . 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 832 . . 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 714 . 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 285 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4394  (class class class)co 6234   RRcr 9441    <_ cle 9579    - cmin 9761   [,]cicc 11503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-icc 11507
This theorem is referenced by:  iccshftli  11628  iccf1o  11635
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