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Theorem iccshftl 11542
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 457 . . . . 5  |-  ( ( X  e.  RR  /\  R  e.  RR )  ->  X  e.  RR )
2 resubcl 9788 . . . . 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 466 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( X  e.  RR  /\  R  e.  RR ) )  -> 
( X  e.  RR  <->  ( X  -  R )  e.  RR ) )
5 lesub1 9948 . . . . . 6  |-  ( ( A  e.  RR  /\  X  e.  RR  /\  R  e.  RR )  ->  ( A  <_  X  <->  ( A  -  R )  <_  ( X  -  R )
) )
653expb 1189 . . . . 5  |-  ( ( A  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( A  <_  X  <->  ( A  -  R )  <_  ( X  -  R )
) )
76adantlr 714 . . . 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 4410 . . . 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 9948 . . . . . . 7  |-  ( ( X  e.  RR  /\  B  e.  RR  /\  R  e.  RR )  ->  ( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R )
) )
12113expb 1189 . . . . . 6  |-  ( ( X  e.  RR  /\  ( B  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R )
) )
1312an12s 799 . . . . 5  |-  ( ( B  e.  RR  /\  ( X  e.  RR  /\  R  e.  RR ) )  ->  ( X  <_  B  <->  ( X  -  R )  <_  ( B  -  R )
) )
1413adantll 713 . . . 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 4411 . . . 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 1290 . 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 11475 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
2019adantr 465 . 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 9788 . . . . . 6  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  ( A  -  R
)  e.  RR )
228, 21syl5eqelr 2547 . . . . 5  |-  ( ( A  e.  RR  /\  R  e.  RR )  ->  C  e.  RR )
23 resubcl 9788 . . . . . 6  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  ( B  -  R
)  e.  RR )
2415, 23syl5eqelr 2547 . . . . 5  |-  ( ( B  e.  RR  /\  R  e.  RR )  ->  D  e.  RR )
25 elicc2 11475 . . . . 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 477 . . . 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 827 . . 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 715 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4403  (class class class)co 6203   RRcr 9396    <_ cle 9534    - cmin 9710   [,]cicc 11418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-icc 11422
This theorem is referenced by:  iccshftli  11543  iccf1o  11550
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