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Theorem iccneg 11020
Description: Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
Assertion
Ref Expression
iccneg  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A [,] B )  <->  -u C  e.  ( -u B [,] -u A ) ) )

Proof of Theorem iccneg
StepHypRef Expression
1 renegcl 9366 . . . . 5  |-  ( C  e.  RR  ->  -u C  e.  RR )
2 ax-1 5 . . . . 5  |-  ( C  e.  RR  ->  ( -u C  e.  RR  ->  C  e.  RR ) )
31, 2impbid2 197 . . . 4  |-  ( C  e.  RR  ->  ( C  e.  RR  <->  -u C  e.  RR ) )
433ad2ant3 981 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  RR  <->  -u C  e.  RR ) )
5 ancom 439 . . . 4  |-  ( ( C  <_  B  /\  A  <_  C )  <->  ( A  <_  C  /\  C  <_  B ) )
6 leneg 9533 . . . . . . 7  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  <_  B  <->  -u B  <_  -u C ) )
76ancoms 441 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  <_  B  <->  -u B  <_  -u C ) )
873adant1 976 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <_  B  <->  -u B  <_  -u C ) )
9 leneg 9533 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  C  <->  -u C  <_  -u A ) )
1093adant2 977 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  C  <->  -u C  <_  -u A ) )
118, 10anbi12d 693 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  <_  B  /\  A  <_  C )  <-> 
( -u B  <_  -u C  /\  -u C  <_  -u A
) ) )
125, 11syl5bbr 252 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <_  C  /\  C  <_  B )  <-> 
( -u B  <_  -u C  /\  -u C  <_  -u A
) ) )
134, 12anbi12d 693 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  e.  RR  /\  ( A  <_  C  /\  C  <_  B ) )  <->  ( -u C  e.  RR  /\  ( -u B  <_  -u C  /\  -u C  <_ 
-u A ) ) ) )
14 elicc2 10977 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A [,] B )  <-> 
( C  e.  RR  /\  A  <_  C  /\  C  <_  B ) ) )
15143adant3 978 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR  /\  A  <_  C  /\  C  <_  B
) ) )
16 3anass 941 . . 3  |-  ( ( C  e.  RR  /\  A  <_  C  /\  C  <_  B )  <->  ( C  e.  RR  /\  ( A  <_  C  /\  C  <_  B ) ) )
1715, 16syl6bb 254 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR  /\  ( A  <_  C  /\  C  <_  B ) ) ) )
18 renegcl 9366 . . . . 5  |-  ( B  e.  RR  ->  -u B  e.  RR )
19 renegcl 9366 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
20 elicc2 10977 . . . . 5  |-  ( (
-u B  e.  RR  /\  -u A  e.  RR )  ->  ( -u C  e.  ( -u B [,] -u A )  <->  ( -u C  e.  RR  /\  -u B  <_ 
-u C  /\  -u C  <_ 
-u A ) ) )
2118, 19, 20syl2anr 466 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u C  e.  ( -u B [,] -u A )  <->  ( -u C  e.  RR  /\  -u B  <_ 
-u C  /\  -u C  <_ 
-u A ) ) )
22213adant3 978 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -u C  e.  ( -u B [,] -u A )  <->  ( -u C  e.  RR  /\  -u B  <_ 
-u C  /\  -u C  <_ 
-u A ) ) )
23 3anass 941 . . 3  |-  ( (
-u C  e.  RR  /\  -u B  <_  -u C  /\  -u C  <_  -u A
)  <->  ( -u C  e.  RR  /\  ( -u B  <_  -u C  /\  -u C  <_ 
-u A ) ) )
2422, 23syl6bb 254 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -u C  e.  ( -u B [,] -u A )  <->  ( -u C  e.  RR  /\  ( -u B  <_  -u C  /\  -u C  <_ 
-u A ) ) ) )
2513, 17, 243bitr4d 278 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A [,] B )  <->  -u C  e.  ( -u B [,] -u A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726   class class class wbr 4214  (class class class)co 6083   RRcr 8991    <_ cle 9123   -ucneg 9294   [,]cicc 10921
This theorem is referenced by:  xrhmeo  18973  dvivth  19896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-icc 10925
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