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Theorem elico2 11698
Description: Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)
Assertion
Ref Expression
elico2  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( C  e.  ( A [,) B )  <-> 
( C  e.  RR  /\  A  <_  C  /\  C  <  B ) ) )

Proof of Theorem elico2
StepHypRef Expression
1 rexr 9686 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
2 elico1 11679 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,) B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B
) ) )
31, 2sylan 474 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( C  e.  ( A [,) B )  <-> 
( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) ) )
4 mnfxr 11414 . . . . . . . 8  |- -oo  e.  RR*
54a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  -> -oo  e.  RR* )
61ad2antrr 732 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  A  e.  RR* )
7 simpr1 1014 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  e.  RR* )
8 mnflt 11425 . . . . . . . 8  |-  ( A  e.  RR  -> -oo  <  A )
98ad2antrr 732 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  -> -oo  <  A )
10 simpr2 1015 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  A  <_  C
)
115, 6, 7, 9, 10xrltletrd 11458 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  -> -oo  <  C )
12 simplr 762 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  B  e.  RR* )
13 pnfxr 11412 . . . . . . . 8  |- +oo  e.  RR*
1413a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  -> +oo  e.  RR* )
15 simpr3 1016 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  <  B
)
16 pnfge 11432 . . . . . . . 8  |-  ( B  e.  RR*  ->  B  <_ +oo )
1716ad2antlr 733 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  B  <_ +oo )
187, 12, 14, 15, 17xrltletrd 11458 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  < +oo )
19 xrrebnd 11463 . . . . . . 7  |-  ( C  e.  RR*  ->  ( C  e.  RR  <->  ( -oo  <  C  /\  C  < +oo ) ) )
207, 19syl 17 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  ( C  e.  RR  <->  ( -oo  <  C  /\  C  < +oo ) ) )
2111, 18, 20mpbir2and 933 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  C  e.  RR )
2221, 10, 153jca 1188 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  A  <_  C  /\  C  <  B ) )  ->  ( C  e.  RR  /\  A  <_  C  /\  C  <  B
) )
2322ex 436 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( C  e. 
RR*  /\  A  <_  C  /\  C  <  B
)  ->  ( C  e.  RR  /\  A  <_  C  /\  C  <  B
) ) )
24 rexr 9686 . . . 4  |-  ( C  e.  RR  ->  C  e.  RR* )
25243anim1i 1194 . . 3  |-  ( ( C  e.  RR  /\  A  <_  C  /\  C  <  B )  ->  ( C  e.  RR*  /\  A  <_  C  /\  C  < 
B ) )
2623, 25impbid1 207 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( C  e. 
RR*  /\  A  <_  C  /\  C  <  B
)  <->  ( C  e.  RR  /\  A  <_  C  /\  C  <  B
) ) )
273, 26bitrd 257 1  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( C  e.  ( A [,) B )  <-> 
( C  e.  RR  /\  A  <_  C  /\  C  <  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    e. wcel 1887   class class class wbr 4402  (class class class)co 6290   RRcr 9538   +oocpnf 9672   -oocmnf 9673   RR*cxr 9674    < clt 9675    <_ cle 9676   [,)cico 11637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-pre-lttri 9613  ax-pre-lttrn 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-ico 11641
This theorem is referenced by:  icossre  11715  elicopnf  11730  icoshft  11754  muladdmodid  12137  icodiamlt  13497  fprodge0  14047  fprodge1  14049  rge0srg  19038  metustexhalf  21571  cnbl0  21794  icoopnst  21967  iocopnst  21968  icopnfcnv  21970  icopnfhmeo  21971  iccpnfcnv  21972  psercnlem2  23379  psercnlem1  23380  psercn  23381  abelth  23396  tanord1  23486  tanord  23487  efopnlem1  23601  logtayl  23605  rlimcnp  23891  rlimcnp2  23892  dchrvmasumlem2  24336  dchrvmasumiflem1  24339  pntlemb  24435  pnt  24452  ubico  28357  xrge0slmod  28607  voliune  29052  volfiniune  29053  dya2icoseg  29099  sibfinima  29172  relowlpssretop  31767  tan2h  31937  itg2addnclem2  31994  modelico  35665  binomcxplemdvbinom  36702  binomcxplemcvg  36703  binomcxplemnotnn0  36705  limciccioolb  37701  fourierdlem32  38002  fourierdlem43  38014  fourierdlem63  38033  fourierdlem79  38049  fouriersw  38095  expnegico01  40368  dignnld  40467
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