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Theorem qextltlem 11390
Description: Lemma for qextlt 11391 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 11388 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
213expia 1193 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
3 simprl 755 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  A  <  x )
4 simplll 757 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  A  e.  RR* )
5 qre 11176 . . . . . . . . . . . 12  |-  ( x  e.  QQ  ->  x  e.  RR )
65rexrd 9632 . . . . . . . . . . 11  |-  ( x  e.  QQ  ->  x  e.  RR* )
76ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  x  e.  RR* )
8 xrltnle 9642 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  x  e.  RR* )  ->  ( A  <  x  <->  -.  x  <_  A ) )
94, 7, 8syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( A  <  x  <->  -.  x  <_  A )
)
103, 9mpbid 210 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  x  <_  A )
11 xrltle 11344 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  A  e.  RR* )  ->  (
x  <  A  ->  x  <_  A ) )
127, 4, 11syl2anc 661 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( x  <  A  ->  x  <_  A )
)
1310, 12mtod 177 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  x  <  A )
14 simprr 756 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  x  <  B )
1513, 142thd 240 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( -.  x  < 
A  <->  x  <  B ) )
16 nbbn 358 . . . . . 6  |-  ( ( -.  x  <  A  <->  x  <  B )  <->  -.  (
x  <  A  <->  x  <  B ) )
1715, 16sylib 196 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  ( x  <  A  <->  x  <  B ) )
18 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  B  e.  RR* )
19 xrltle 11344 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  B  e.  RR* )  ->  (
x  <  B  ->  x  <_  B ) )
207, 18, 19syl2anc 661 . . . . . . . 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 240 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( -.  x  <_  A 
<->  x  <_  B )
)
23 nbbn 358 . . . . . 6  |-  ( ( -.  x  <_  A  <->  x  <_  B )  <->  -.  (
x  <_  A  <->  x  <_  B ) )
2422, 23sylib 196 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  ->  -.  ( x  <_  A  <->  x  <_  B ) )
2517, 24jca 532 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  /\  ( A  <  x  /\  x  <  B ) )  -> 
( -.  ( x  <  A  <->  x  <  B )  /\  -.  (
x  <_  A  <->  x  <_  B ) ) )
2625ex 434 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  x  e.  QQ )  ->  ( ( A  <  x  /\  x  <  B )  ->  ( -.  ( x  <  A  <->  x  <  B )  /\  -.  ( x  <_  A  <->  x  <_  B ) ) ) )
2726reximdva 2931 . 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 44 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1762   E.wrex 2808   class class class wbr 4440   RR*cxr 9616    < clt 9617    <_ cle 9618   QQcq 11171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-q 11172
This theorem is referenced by:  qextlt  11391  qextle  11392
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