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Theorem qbtwnxr 10527
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 10458 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
2 elxr 10458 . . . . 5  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = 
+oo  \/  B  =  -oo ) )
3 qbtwnre 10526 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
433expia 1153 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
5 simpl 443 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  A  e.  RR )
6 peano2re 8985 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
76adantr 451 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( A  +  1 )  e.  RR )
8 ltp1 9594 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
98adantr 451 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  A  <  ( A  + 
1 ) )
10 qbtwnre 10526 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( A  +  1
)  e.  RR  /\  A  <  ( A  + 
1 ) )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
115, 7, 9, 10syl3anc 1182 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
12 qre 10321 . . . . . . . . . . . . . 14  |-  ( x  e.  QQ  ->  x  e.  RR )
13 ltpnf 10463 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  x  <  +oo )
1412, 13syl 15 . . . . . . . . . . . . 13  |-  ( x  e.  QQ  ->  x  <  +oo )
1514adantl 452 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  =  +oo )  /\  x  e.  QQ )  ->  x  <  +oo )
16 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  =  +oo )  /\  x  e.  QQ )  ->  B  =  +oo )
1715, 16breqtrrd 4049 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  =  +oo )  /\  x  e.  QQ )  ->  x  <  B
)
1817a1d 22 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  =  +oo )  /\  x  e.  QQ )  ->  ( x  < 
( A  +  1 )  ->  x  <  B ) )
1918anim2d 548 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  =  +oo )  /\  x  e.  QQ )  ->  ( ( A  <  x  /\  x  <  ( A  +  1 ) )  ->  ( A  <  x  /\  x  <  B ) ) )
2019reximdva 2655 . . . . . . . 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 14 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
2221a1d 22 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
23 rexr 8877 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
24 breq2 4027 . . . . . . . . 9  |-  ( B  =  -oo  ->  ( A  <  B  <->  A  <  -oo ) )
2524adantl 452 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  =  -oo )  ->  ( A  <  B  <->  A  <  -oo ) )
26 nltmnf 10468 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  -.  A  <  -oo )
2726adantr 451 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  =  -oo )  ->  -.  A  <  -oo )
2827pm2.21d 98 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  =  -oo )  ->  ( A  <  -oo  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
2925, 28sylbid 206 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  =  -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
3023, 29sylan 457 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
314, 22, 303jaodan 1248 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
322, 31sylan2b 461 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
33 breq1 4026 . . . . . 6  |-  ( A  =  +oo  ->  ( A  <  B  <->  +oo  <  B
) )
3433adantr 451 . . . . 5  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
( A  <  B  <->  +oo 
<  B ) )
35 pnfnlt 10467 . . . . . . 7  |-  ( B  e.  RR*  ->  -.  +oo  <  B )
3635adantl 452 . . . . . 6  |-  ( ( A  =  +oo  /\  B  e.  RR* )  ->  -.  +oo  <  B )
3736pm2.21d 98 . . . . 5  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
(  +oo  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
3834, 37sylbid 206 . . . 4  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
39 peano2rem 9113 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
4039adantl 452 . . . . . . . . 9  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( B  -  1 )  e.  RR )
41 simpr 447 . . . . . . . . 9  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  B  e.  RR )
42 ltm1 9596 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  <  B )
4342adantl 452 . . . . . . . . 9  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( B  -  1 )  <  B )
44 qbtwnre 10526 . . . . . . . . 9  |-  ( ( ( B  -  1 )  e.  RR  /\  B  e.  RR  /\  ( B  -  1 )  <  B )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
4540, 41, 43, 44syl3anc 1182 . . . . . . . 8  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
46 simpll 730 . . . . . . . . . . . 12  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  A  =  -oo )
4712adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  x  e.  RR )
48 mnflt 10464 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  -oo  <  x )
4947, 48syl 15 . . . . . . . . . . . 12  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  -oo  <  x )
5046, 49eqbrtrd 4043 . . . . . . . . . . 11  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  A  <  x
)
5150a1d 22 . . . . . . . . . 10  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  ( ( B  -  1 )  < 
x  ->  A  <  x ) )
5251anim1d 547 . . . . . . . . 9  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  ( ( ( B  -  1 )  <  x  /\  x  <  B )  ->  ( A  <  x  /\  x  <  B ) ) )
5352reximdva 2655 . . . . . . . 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 14 . . . . . . 7  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
5554a1d 22 . . . . . 6  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
56 1re 8837 . . . . . . . . . 10  |-  1  e.  RR
57 mnflt 10464 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  -oo  <  1 )
5856, 57ax-mp 8 . . . . . . . . 9  |-  -oo  <  1
59 breq1 4026 . . . . . . . . 9  |-  ( A  =  -oo  ->  ( A  <  1  <->  -oo  <  1
) )
6058, 59mpbiri 224 . . . . . . . 8  |-  ( A  =  -oo  ->  A  <  1 )
61 ltpnf 10463 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  1  <  +oo )
6256, 61ax-mp 8 . . . . . . . . 9  |-  1  <  +oo
63 breq2 4027 . . . . . . . . 9  |-  ( B  =  +oo  ->  (
1  <  B  <->  1  <  +oo ) )
6462, 63mpbiri 224 . . . . . . . 8  |-  ( B  =  +oo  ->  1  <  B )
65 1z 10053 . . . . . . . . . 10  |-  1  e.  ZZ
66 zq 10322 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
6765, 66ax-mp 8 . . . . . . . . 9  |-  1  e.  QQ
68 breq2 4027 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( A  <  x  <->  A  <  1 ) )
69 breq1 4026 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  <  B  <->  1  <  B ) )
7068, 69anbi12d 691 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( A  <  x  /\  x  <  B )  <-> 
( A  <  1  /\  1  <  B ) ) )
7170rspcev 2884 . . . . . . . . 9  |-  ( ( 1  e.  QQ  /\  ( A  <  1  /\  1  <  B ) )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
7267, 71mpan 651 . . . . . . . 8  |-  ( ( A  <  1  /\  1  <  B )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
7360, 64, 72syl2an 463 . . . . . . 7  |-  ( ( A  =  -oo  /\  B  =  +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
7473a1d 22 . . . . . 6  |-  ( ( A  =  -oo  /\  B  =  +oo )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
75 3mix3 1126 . . . . . . . 8  |-  ( A  =  -oo  ->  ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo ) )
7675, 1sylibr 203 . . . . . . 7  |-  ( A  =  -oo  ->  A  e.  RR* )
7776, 29sylan 457 . . . . . 6  |-  ( ( A  =  -oo  /\  B  =  -oo )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
7855, 74, 773jaodan 1248 . . . . 5  |-  ( ( A  =  -oo  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
792, 78sylan2b 461 . . . 4  |-  ( ( A  =  -oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
8032, 38, 793jaoian 1247 . . 3  |-  ( ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
811, 80sylanb 458 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
82813impia 1148 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023  (class class class)co 5858   RRcr 8736   1c1 8738    + caddc 8740    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    - cmin 9037   ZZcz 10024   QQcq 10316
This theorem is referenced by:  qextltlem  10529  xralrple  10532  ixxub  10677  ixxlb  10678  ioo0  10681  ico0  10702  ioc0  10703  blss  17972  blcld  18051  qdensere  18279  tgqioo  18306  dvlip2  19342  lhop2  19362  itgsubst  19396
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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