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Theorem elico2 10094
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 8153 . . 3 |- ( A e. RR -> A e. RR* )
2 elico1 10080 . . 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 8164 . . . . . . . 8 |- -oo e. RR*
54a1i 9 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo e. RR* )
61ad2antrr 488 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> A e. RR* )
7 simpr1 1006 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C e. RR* )
8 mnflt 9940 . . . . . . . 8 |- ( A e. RR -> -oo < A )
98ad2antrr 488 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo < A )
10 simpr2 1007 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> A <_ C )
115, 6, 7, 9, 10xrltletrd 9968 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> -oo < C )
12 simplr 528 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> B e. RR* )
13 pnfxr 8160 . . . . . . . 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 1008 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C < B )
16 pnfge 9946 . . . . . . . 8 |- ( B e. RR* -> B <_ +oo )
1716ad2antlr 489 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> B <_ +oo )
187, 12, 14, 15, 17xrltletrd 9968 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C < +oo )
19 xrrebnd 9976 . . . . . . 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 947 . . . . 5 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR* /\ A <_ C /\ C < B ) ) -> C e. RR )
2221, 10, 153jca 1180 . . . 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 8153 . . . 4 |- ( C e. RR -> C e. RR* )
25243anim1i 1188 . . 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
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   RRcr 7959   +oocpnf 8139   -oocmnf 8140   RR*cxr 8141   < clt 8142   <_ cle 8143   [,)cico 10047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074
This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-ico 10051
This theorem is referenced by:  icossre  10111  elicopnf  10126  icoshft  10147  modqelico  10516  mulqaddmodid  10546  modqmuladdim  10549  addmodid  10554  icodiamlt  11606  fprodge0  12063  fprodge1  12065  cnbl0  15121  cosq34lt1  15437  cos02pilt1  15438
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