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Theorem elico2 10349
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 8371 . . 3 |- ( A e. RR -> A e. RR* )
2 elico1 10335 . . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A [,) B ) <-> ( C e. RR* /\ A <_ C /\ C < B ) ) )
31, 2sylan 283 . 2 |- ( ( A e. RR /\ B e. RR* ) -> ( C e. ( A [,) B ) <-> ( C e. RR* /\ A <_ C /\ C < B ) ) )
4 mnfxr 8382 . . . . . . . 8 |- -oo e. RR*
54a1i 9 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo e. RR* )
61ad2antrr 492 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> A e. RR* )
7 simpr1 1034 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C e. RR* )
8 mnflt 10195 . . . . . . . 8 |- ( A e. RR -> -oo < A )
98ad2antrr 492 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo < A )
10 simpr2 1035 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> A <_ C )
115, 6, 7, 9, 10xrltletrd 10223 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo < C )
12 simplr 533 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> B e. RR* )
13 pnfxr 8378 . . . . . . . 8 |- +oo e. RR*
1413a1i 9 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> +oo e. RR* )
15 simpr3 1036 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C < B )
16 pnfge 10201 . . . . . . . 8 |- ( B e. RR* -> B <_ +oo )
1716ad2antlr 493 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> B <_ +oo )
187, 12, 14, 15, 17xrltletrd 10223 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C < +oo )
19 xrrebnd 10231 . . . . . . 7 |- ( C e. RR* -> ( C e. RR <-> ( -oo < C /\ C < +oo ) ) )
207, 19syl 14 . . . . . 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 1208 . . . 4 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> ( C e. RR /\ A <_ C /\ C < B ) )
2322ex 115 . . 3 |- ( ( A e. RR /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C < B ) -> ( C e. RR /\ A <_ C /\ C < B ) ) )
24 rexr 8371 . . . 4 |- ( C e. RR -> C e. RR* )
25243anim1i 1216 . . 3 |- ( ( C e. RR /\ A <_ C /\ C < B ) -> ( C e. RR* /\ A <_ C /\ C < B ) )
2623, 25impbid1 142 . 2 |- ( ( A e. RR /\ B e. RR* ) -> ( ( C e. RR* /\ A <_ C /\ C < B ) <-> ( C e. RR /\ A <_ C /\ C < B ) ) )
273, 26bitrd 188 1 |- ( ( A e. RR /\ B e. RR* ) -> ( C e. ( A [,) B ) <-> ( C e. RR /\ A <_ C /\ C < B ) ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   RRcr 8178   +oocpnf 8357   -oocmnf 8358   RR*cxr 8359   < clt 8360   <_ cle 8361   [,)cico 10302
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293
This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-ico 10306
This theorem is used by:  icossre  10366  elicopnf  10381  icoshft  10402  modqelico  10784  mulqaddmodid  10814  modqmuladdim  10817  addmodid  10822  icodiamlt  11961  fprodge0  12420  fprodge1  12422  cnbl0  15684  cosq34lt1  16001  cos02pilt1  16002  repiecelem  17172  repiecele0  17173  repiecege0  17174
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