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Theorem qextltlem 10819
Description: Lemma for qextlt 10820 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 10817 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
213expia 1156 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
3 simprl 734 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  A  <  x )
4 simplll 736 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  A  e.  RR* )
5 qre 10610 . . . . . . . . . . . 12  |-  ( x  e.  QQ  ->  x  e.  RR )
65rexrd 9165 . . . . . . . . . . 11  |-  ( x  e.  QQ  ->  x  e.  RR* )
76ad2antlr 709 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  x  e.  RR* )
8 xrltnle 9175 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <  x  <->  -.  x  <_  A ) )
94, 7, 8syl2anc 644 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( A  <  x  <->  -.  x  <_  A )
)
103, 9mpbid 203 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  x  <_  A )
11 xrltle 10773 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
x  <  A  ->  x  <_  A ) )
127, 4, 11syl2anc 644 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( x  <  A  ->  x  <_  A )
)
1310, 12mtod 171 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  x  <  A )
14 simprr 735 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  x  <  B )
1513, 142thd 233 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( -.  x  < 
A  <->  x  <  B ) )
16 nbbn 349 . . . . . 6  |-  ( ( -.  x  <  A  <->  x  <  B )  <->  -.  (
x  <  A  <->  x  <  B ) )
1715, 16sylib 190 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  ( x  <  A  <->  x  <  B ) )
18 simpllr 737 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  B  e.  RR* )
19 xrltle 10773 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  B  e.  RR* )  ->  (
x  <  B  ->  x  <_  B ) )
207, 18, 19syl2anc 644 . . . . . . . 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 233 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( -.  x  <_  A 
<->  x  <_  B )
)
23 nbbn 349 . . . . . 6  |-  ( ( -.  x  <_  A  <->  x  <_  B )  <->  -.  (
x  <_  A  <->  x  <_  B ) )
2422, 23sylib 190 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  ( x  <_  A  <->  x  <_  B ) )
2517, 24jca 520 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( -.  ( x  <  A  <->  x  <  B )  /\  -.  (
x  <_  A  <->  x  <_  B ) ) )
2625ex 425 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  ->  ( ( A  <  x  /\  x  <  B )  ->  ( -.  ( x  <  A  <->  x  <  B )  /\  -.  ( x  <_  A  <->  x  <_  B ) ) ) )
2726reximdva 2824 . 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 43 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 178    /\ wa 360    e. wcel 1727   E.wrex 2712   class class class wbr 4237   RR*cxr 9150    < clt 9151    <_ cle 9152   QQcq 10605
This theorem is referenced by:  qextlt  10820  qextle  10821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-sup 7475  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-n0 10253  df-z 10314  df-uz 10520  df-q 10606
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