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Theorem qextltlem 10529
Description: Lemma for qextlt 10530 and qextle . (Contributed by Mario Carneiro, 3-Oct-2014.)
Assertion
Ref Expression
qextltlem  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( -.  (
x  <  A  <->  x  <  B )  /\  -.  (
x  <_  A  <->  x  <_  B ) ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem qextltlem
StepHypRef Expression
1 qbtwnxr 10527 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
213expia 1153 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
3 simprl 732 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  A  <  x )
4 simplll 734 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  A  e.  RR* )
5 qre 10321 . . . . . . . . . . . 12  |-  ( x  e.  QQ  ->  x  e.  RR )
65rexrd 8881 . . . . . . . . . . 11  |-  ( x  e.  QQ  ->  x  e.  RR* )
76ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  x  e.  RR* )
8 xrltnle 8891 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <  x  <->  -.  x  <_  A ) )
94, 7, 8syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( A  <  x  <->  -.  x  <_  A )
)
103, 9mpbid 201 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  x  <_  A )
11 xrltle 10483 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
x  <  A  ->  x  <_  A ) )
127, 4, 11syl2anc 642 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( x  <  A  ->  x  <_  A )
)
1310, 12mtod 168 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  x  <  A )
14 simprr 733 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  x  <  B )
1513, 142thd 231 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( -.  x  < 
A  <->  x  <  B ) )
16 nbbn 347 . . . . . 6  |-  ( ( -.  x  <  A  <->  x  <  B )  <->  -.  (
x  <  A  <->  x  <  B ) )
1715, 16sylib 188 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  ( x  <  A  <->  x  <  B ) )
18 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  B  e.  RR* )
19 xrltle 10483 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  B  e.  RR* )  ->  (
x  <  B  ->  x  <_  B ) )
207, 18, 19syl2anc 642 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( x  <  B  ->  x  <_  B )
)
2114, 20mpd 14 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  x  <_  B )
2210, 212thd 231 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( -.  x  <_  A 
<->  x  <_  B )
)
23 nbbn 347 . . . . . 6  |-  ( ( -.  x  <_  A  <->  x  <_  B )  <->  -.  (
x  <_  A  <->  x  <_  B ) )
2422, 23sylib 188 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  ( x  <_  A  <->  x  <_  B ) )
2517, 24jca 518 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( -.  ( x  <  A  <->  x  <  B )  /\  -.  (
x  <_  A  <->  x  <_  B ) ) )
2625ex 423 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  ->  ( ( A  <  x  /\  x  <  B )  ->  ( -.  ( x  <  A  <->  x  <  B )  /\  -.  ( x  <_  A  <->  x  <_  B ) ) ) )
2726reximdva 2655 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  QQ  ( A  <  x  /\  x  <  B )  ->  E. x  e.  QQ  ( -.  ( x  <  A  <->  x  <  B )  /\  -.  ( x  <_  A  <->  x  <_  B ) ) ) )
282, 27syld 40 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( -.  (
x  <  A  <->  x  <  B )  /\  -.  (
x  <_  A  <->  x  <_  B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   E.wrex 2544   class class class wbr 4023   RR*cxr 8866    < clt 8867    <_ cle 8868   QQcq 10316
This theorem is referenced by:  qextlt  10530  qextle  10531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317
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