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Theorem qextltlem 10752
Description: Lemma for qextlt 10753 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 10750 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
213expia 1155 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
3 simprl 733 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  A  <  x )
4 simplll 735 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  A  e.  RR* )
5 qre 10543 . . . . . . . . . . . 12  |-  ( x  e.  QQ  ->  x  e.  RR )
65rexrd 9098 . . . . . . . . . . 11  |-  ( x  e.  QQ  ->  x  e.  RR* )
76ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  x  e.  RR* )
8 xrltnle 9108 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <  x  <->  -.  x  <_  A ) )
94, 7, 8syl2anc 643 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( A  <  x  <->  -.  x  <_  A )
)
103, 9mpbid 202 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  x  <_  A )
11 xrltle 10706 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
x  <  A  ->  x  <_  A ) )
127, 4, 11syl2anc 643 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( x  <  A  ->  x  <_  A )
)
1310, 12mtod 170 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  x  <  A )
14 simprr 734 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  x  <  B )
1513, 142thd 232 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( -.  x  < 
A  <->  x  <  B ) )
16 nbbn 348 . . . . . 6  |-  ( ( -.  x  <  A  <->  x  <  B )  <->  -.  (
x  <  A  <->  x  <  B ) )
1715, 16sylib 189 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  ( x  <  A  <->  x  <  B ) )
18 simpllr 736 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  B  e.  RR* )
19 xrltle 10706 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  B  e.  RR* )  ->  (
x  <  B  ->  x  <_  B ) )
207, 18, 19syl2anc 643 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( x  <  B  ->  x  <_  B )
)
2114, 20mpd 15 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  x  <_  B )
2210, 212thd 232 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( -.  x  <_  A 
<->  x  <_  B )
)
23 nbbn 348 . . . . . 6  |-  ( ( -.  x  <_  A  <->  x  <_  B )  <->  -.  (
x  <_  A  <->  x  <_  B ) )
2422, 23sylib 189 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  ( x  <_  A  <->  x  <_  B ) )
2517, 24jca 519 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( -.  ( x  <  A  <->  x  <  B )  /\  -.  (
x  <_  A  <->  x  <_  B ) ) )
2625ex 424 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  ->  ( ( A  <  x  /\  x  <  B )  ->  ( -.  ( x  <  A  <->  x  <  B )  /\  -.  ( x  <_  A  <->  x  <_  B ) ) ) )
2726reximdva 2786 . 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 42 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 177    /\ wa 359    e. wcel 1721   E.wrex 2675   class class class wbr 4180   RR*cxr 9083    < clt 9084    <_ cle 9085   QQcq 10538
This theorem is referenced by:  qextlt  10753  qextle  10754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539
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