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Theorem iooneg 11017
Description: Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
Assertion
Ref Expression
iooneg  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A (,) B )  <->  -u C  e.  ( -u B (,) -u A ) ) )

Proof of Theorem iooneg
StepHypRef Expression
1 ltneg 9528 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  -u C  <  -u A
) )
213adant2 976 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  -u C  <  -u A ) )
3 ltneg 9528 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C
) )
43ancoms 440 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C
) )
543adant1 975 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C ) )
62, 5anbi12d 692 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  C  /\  C  <  B )  <-> 
( -u C  <  -u A  /\  -u B  <  -u C
) ) )
7 ancom 438 . . 3  |-  ( (
-u C  <  -u A  /\  -u B  <  -u C
)  <->  ( -u B  <  -u C  /\  -u C  <  -u A ) )
86, 7syl6bb 253 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  C  /\  C  <  B )  <-> 
( -u B  <  -u C  /\  -u C  <  -u A
) ) )
9 rexr 9130 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
10 rexr 9130 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
11 rexr 9130 . . 3  |-  ( C  e.  RR  ->  C  e.  RR* )
12 elioo5 10968 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C  /\  C  < 
B ) ) )
139, 10, 11, 12syl3an 1226 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C  /\  C  < 
B ) ) )
14 renegcl 9364 . . . 4  |-  ( B  e.  RR  ->  -u B  e.  RR )
15 renegcl 9364 . . . 4  |-  ( A  e.  RR  ->  -u A  e.  RR )
16 renegcl 9364 . . . 4  |-  ( C  e.  RR  ->  -u C  e.  RR )
17 rexr 9130 . . . . 5  |-  ( -u B  e.  RR  ->  -u B  e.  RR* )
18 rexr 9130 . . . . 5  |-  ( -u A  e.  RR  ->  -u A  e.  RR* )
19 rexr 9130 . . . . 5  |-  ( -u C  e.  RR  ->  -u C  e.  RR* )
20 elioo5 10968 . . . . 5  |-  ( (
-u B  e.  RR*  /\  -u A  e.  RR*  /\  -u C  e.  RR* )  ->  ( -u C  e.  ( -u B (,) -u A )  <->  ( -u B  <  -u C  /\  -u C  <  -u A ) ) )
2117, 18, 19, 20syl3an 1226 . . . 4  |-  ( (
-u B  e.  RR  /\  -u A  e.  RR  /\  -u C  e.  RR )  ->  ( -u C  e.  ( -u B (,) -u A )  <->  ( -u B  <  -u C  /\  -u C  <  -u A ) ) )
2214, 15, 16, 21syl3an 1226 . . 3  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( -u C  e.  ( -u B (,) -u A )  <->  ( -u B  <  -u C  /\  -u C  <  -u A ) ) )
23223com12 1157 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -u C  e.  ( -u B (,) -u A )  <->  ( -u B  <  -u C  /\  -u C  <  -u A ) ) )
248, 13, 233bitr4d 277 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 177    /\ wa 359    /\ w3a 936    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   RRcr 8989   RR*cxr 9119    < clt 9120   -ucneg 9292   (,)cioo 10916
This theorem is referenced by:  lhop2  19899  asinsin  20732  atanlogsub  20756  atanbnd  20766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-ioo 10920
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