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Theorem elico2 12237
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 10085 . . 3 |- ( A e. RR -> A e. RR* )
2 elico1 12218 . . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,) B ) <-> ( C e. RR* /\ A <_ C /\ C < B ) ) )
31, 2sylan 488 . 2 |- ( ( A e. RR /\ B e. RR* ) -> ( C e. ( A [,) B ) <-> ( C e. RR* /\ A <_ C /\ C < B ) ) )
4 mnfxr 10096 . . . . . . . 8 |- -oo e. RR*
54a1i 11 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo e. RR* )
61ad2antrr 762 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> A e. RR* )
7 simpr1 1067 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C e. RR* )
8 mnflt 11957 . . . . . . . 8 |- ( A e. RR -> -oo < A )
98ad2antrr 762 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo < A )
10 simpr2 1068 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> A <_ C )
115, 6, 7, 9, 10xrltletrd 11992 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo < C )
12 simplr 792 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> B e. RR* )
13 pnfxr 10092 . . . . . . . 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 1069 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C < B )
16 pnfge 11964 . . . . . . . 8 |- ( B e. RR* -> B <_ +oo )
1716ad2antlr 763 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> B <_ +oo )
187, 12, 14, 15, 17xrltletrd 11992 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C < +oo )
19 xrrebnd 11999 . . . . . . 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 957 . . . . 5 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C e. RR )
2221, 10, 153jca 1242 . . . 4 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> ( C e. RR /\ A <_ C /\ C < B ) )
2322ex 450 . . 3 |- ( ( A e. RR /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C < B ) -> ( C e. RR /\ A <_ C /\ C < B ) ) )
24 rexr 10085 . . . 4 |- ( C e. RR -> C e. RR* )
25243anim1i 1248 . . 3 |- ( ( C e. RR /\ A <_ C /\ C < B ) -> ( C e. RR* /\ A <_ C /\ C < B ) )
2623, 25impbid1 215 . 2 |- ( ( A e. RR /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C < B ) <-> ( C e. RR /\ A <_ C /\ C < B ) ) )
273, 26bitrd 268 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 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   RRcr 9935   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   [,)cico 12177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ico 12181
This theorem is referenced by:  icossre  12254  elicopnf  12269  icoshft  12294  modelico  12680  muladdmodid  12710  icodiamlt  14174  fprodge0  14724  fprodge1  14726  rge0srg  19817  metustexhalf  22361  cnbl0  22577  icoopnst  22738  iocopnst  22739  icopnfcnv  22741  icopnfhmeo  22742  iccpnfcnv  22743  psercnlem2  24178  psercnlem1  24179  psercn  24180  abelth  24195  tanord1  24283  tanord  24284  efopnlem1  24402  logtayl  24406  rlimcnp  24692  rlimcnp2  24693  dchrvmasumlem2  25187  dchrvmasumiflem1  25190  pntlemb  25286  pnt  25303  ubico  29537  xrge0slmod  29844  voliune  30292  volfiniune  30293  dya2icoseg  30339  sibfinima  30401  relowlpssretop  33212  tan2h  33401  itg2addnclem2  33462  binomcxplemdvbinom  38552  binomcxplemcvg  38553  binomcxplemnotnn0  38555  limciccioolb  39853  fourierdlem32  40356  fourierdlem43  40367  fourierdlem63  40386  fourierdlem79  40402  fouriersw  40448  expnegico01  42308  dignnld  42397
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