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Theorem qbtwnxr 11452
Description: The rational numbers are dense in  RR*: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
qbtwnxr  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem qbtwnxr
StepHypRef Expression
1 elxr 11378 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11378 . . . . 5  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 qbtwnre 11451 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
433expia 1199 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
5 simpl 455 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  e.  RR )
6 peano2re 9787 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
76adantr 463 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  +  1 )  e.  RR )
8 ltp1 10421 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
98adantr 463 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  <  ( A  +  1 ) )
10 qbtwnre 11451 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( A  +  1
)  e.  RR  /\  A  <  ( A  + 
1 ) )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
115, 7, 9, 10syl3anc 1230 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
12 qre 11232 . . . . . . . . . . . . . 14  |-  ( x  e.  QQ  ->  x  e.  RR )
13 ltpnf 11384 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  x  < +oo )
1412, 13syl 17 . . . . . . . . . . . . 13  |-  ( x  e.  QQ  ->  x  < +oo )
1514adantl 464 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  x  < +oo )
16 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  B  = +oo )
1715, 16breqtrrd 4421 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  x  <  B
)
1817a1d 25 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  ( x  < 
( A  +  1 )  ->  x  <  B ) )
1918anim2d 563 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  ( ( A  <  x  /\  x  <  ( A  +  1 ) )  ->  ( A  <  x  /\  x  <  B ) ) )
2019reximdva 2879 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( E. x  e.  QQ  ( A  < 
x  /\  x  <  ( A  +  1 ) )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
2111, 20mpd 15 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
2221a1d 25 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
23 rexr 9669 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
24 breq2 4399 . . . . . . . . 9  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
2524adantl 464 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
26 nltmnf 11391 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  -.  A  < -oo )
2726adantr 463 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  -.  A  < -oo )
2827pm2.21d 106 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  ( A  < -oo  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
2925, 28sylbid 215 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
3023, 29sylan 469 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
314, 22, 303jaodan 1296 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
322, 31sylan2b 473 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
33 breq1 4398 . . . . . 6  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
3433adantr 463 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
35 pnfnlt 11390 . . . . . . 7  |-  ( B  e.  RR*  ->  -. +oo  <  B )
3635adantl 464 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
3736pm2.21d 106 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( +oo  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
3834, 37sylbid 215 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
39 peano2rem 9922 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
4039adantl 464 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  -  1 )  e.  RR )
41 simpr 459 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  B  e.  RR )
42 ltm1 10423 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  <  B )
4342adantl 464 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  -  1 )  <  B )
44 qbtwnre 11451 . . . . . . . . 9  |-  ( ( ( B  -  1 )  e.  RR  /\  B  e.  RR  /\  ( B  -  1 )  <  B )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
4540, 41, 43, 44syl3anc 1230 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
46 simpll 752 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  A  = -oo )
4712adantl 464 . . . . . . . . . . . . 13  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  x  e.  RR )
48 mnflt 11386 . . . . . . . . . . . . 13  |-  ( x  e.  RR  -> -oo  <  x )
4947, 48syl 17 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  -> -oo  <  x )
5046, 49eqbrtrd 4415 . . . . . . . . . . 11  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  A  <  x
)
5150a1d 25 . . . . . . . . . 10  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  ( ( B  -  1 )  < 
x  ->  A  <  x ) )
5251anim1d 562 . . . . . . . . 9  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  ( ( ( B  -  1 )  <  x  /\  x  <  B )  ->  ( A  <  x  /\  x  <  B ) ) )
5352reximdva 2879 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( E. x  e.  QQ  ( ( B  -  1 )  < 
x  /\  x  <  B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
5445, 53mpd 15 . . . . . . 7  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
5554a1d 25 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
56 1re 9625 . . . . . . . . . 10  |-  1  e.  RR
57 mnflt 11386 . . . . . . . . . 10  |-  ( 1  e.  RR  -> -oo  <  1 )
5856, 57ax-mp 5 . . . . . . . . 9  |- -oo  <  1
59 breq1 4398 . . . . . . . . 9  |-  ( A  = -oo  ->  ( A  <  1  <-> -oo  <  1
) )
6058, 59mpbiri 233 . . . . . . . 8  |-  ( A  = -oo  ->  A  <  1 )
61 ltpnf 11384 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  1  < +oo )
6256, 61ax-mp 5 . . . . . . . . 9  |-  1  < +oo
63 breq2 4399 . . . . . . . . 9  |-  ( B  = +oo  ->  (
1  <  B  <->  1  < +oo ) )
6462, 63mpbiri 233 . . . . . . . 8  |-  ( B  = +oo  ->  1  <  B )
65 1z 10935 . . . . . . . . . 10  |-  1  e.  ZZ
66 zq 11233 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
6765, 66ax-mp 5 . . . . . . . . 9  |-  1  e.  QQ
68 breq2 4399 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( A  <  x  <->  A  <  1 ) )
69 breq1 4398 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  <  B  <->  1  <  B ) )
7068, 69anbi12d 709 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( A  <  x  /\  x  <  B )  <-> 
( A  <  1  /\  1  <  B ) ) )
7170rspcev 3160 . . . . . . . . 9  |-  ( ( 1  e.  QQ  /\  ( A  <  1  /\  1  <  B ) )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
7267, 71mpan 668 . . . . . . . 8  |-  ( ( A  <  1  /\  1  <  B )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
7360, 64, 72syl2an 475 . . . . . . 7  |-  ( ( A  = -oo  /\  B  = +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
7473a1d 25 . . . . . 6  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
75 3mix3 1168 . . . . . . . 8  |-  ( A  = -oo  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
7675, 1sylibr 212 . . . . . . 7  |-  ( A  = -oo  ->  A  e.  RR* )
7776, 29sylan 469 . . . . . 6  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
7855, 74, 773jaodan 1296 . . . . 5  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
792, 78sylan2b 473 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
8032, 38, 793jaoian 1295 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  ( A  < 
B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
811, 80sylanb 470 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
82813impia 1194 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    \/ w3o 973    /\ w3a 974    = wceq 1405    e. wcel 1842   E.wrex 2755   class class class wbr 4395  (class class class)co 6278   RRcr 9521   1c1 9523    + caddc 9525   +oocpnf 9655   -oocmnf 9656   RR*cxr 9657    < clt 9658    - cmin 9841   ZZcz 10905   QQcq 11227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228
This theorem is referenced by:  qextltlem  11454  xralrple  11457  ixxub  11603  ixxlb  11604  ioo0  11607  ico0  11628  ioc0  11629  blssps  21219  blss  21220  blcld  21300  qdensere  21569  tgqioo  21597  dvlip2  22688  lhop2  22708  itgsubst  22742  itg2gt0cn  31443
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