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Theorem qbtwnxr 11162
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 11088 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11088 . . . . 5  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 qbtwnre 11161 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
433expia 1189 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
5 simpl 457 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  e.  RR )
6 peano2re 9534 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
76adantr 465 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  +  1 )  e.  RR )
8 ltp1 10159 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
98adantr 465 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  <  ( A  +  1 ) )
10 qbtwnre 11161 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( A  +  1
)  e.  RR  /\  A  <  ( A  + 
1 ) )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
115, 7, 9, 10syl3anc 1218 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
12 qre 10950 . . . . . . . . . . . . . 14  |-  ( x  e.  QQ  ->  x  e.  RR )
13 ltpnf 11094 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  x  < +oo )
1412, 13syl 16 . . . . . . . . . . . . 13  |-  ( x  e.  QQ  ->  x  < +oo )
1514adantl 466 . . . . . . . . . . . 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 4313 . . . . . . . . . . 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 565 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  ( ( A  <  x  /\  x  <  ( A  +  1 ) )  ->  ( A  <  x  /\  x  <  B ) ) )
2019reximdva 2823 . . . . . . . 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 9421 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
24 breq2 4291 . . . . . . . . 9  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
2524adantl 466 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
26 nltmnf 11101 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  -.  A  < -oo )
2726adantr 465 . . . . . . . . 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 471 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
314, 22, 303jaodan 1284 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
322, 31sylan2b 475 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
33 breq1 4290 . . . . . 6  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
3433adantr 465 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
35 pnfnlt 11100 . . . . . . 7  |-  ( B  e.  RR*  ->  -. +oo  <  B )
3635adantl 466 . . . . . 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 9667 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
4039adantl 466 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  -  1 )  e.  RR )
41 simpr 461 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  B  e.  RR )
42 ltm1 10161 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  <  B )
4342adantl 466 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  -  1 )  <  B )
44 qbtwnre 11161 . . . . . . . . 9  |-  ( ( ( B  -  1 )  e.  RR  /\  B  e.  RR  /\  ( B  -  1 )  <  B )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
4540, 41, 43, 44syl3anc 1218 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
46 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  A  = -oo )
4712adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  x  e.  RR )
48 mnflt 11096 . . . . . . . . . . . . 13  |-  ( x  e.  RR  -> -oo  <  x )
4947, 48syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  -> -oo  <  x )
5046, 49eqbrtrd 4307 . . . . . . . . . . 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 564 . . . . . . . . 9  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  ( ( ( B  -  1 )  <  x  /\  x  <  B )  ->  ( A  <  x  /\  x  <  B ) ) )
5352reximdva 2823 . . . . . . . 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 9377 . . . . . . . . . 10  |-  1  e.  RR
57 mnflt 11096 . . . . . . . . . 10  |-  ( 1  e.  RR  -> -oo  <  1 )
5856, 57ax-mp 5 . . . . . . . . 9  |- -oo  <  1
59 breq1 4290 . . . . . . . . 9  |-  ( A  = -oo  ->  ( A  <  1  <-> -oo  <  1
) )
6058, 59mpbiri 233 . . . . . . . 8  |-  ( A  = -oo  ->  A  <  1 )
61 ltpnf 11094 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  1  < +oo )
6256, 61ax-mp 5 . . . . . . . . 9  |-  1  < +oo
63 breq2 4291 . . . . . . . . 9  |-  ( B  = +oo  ->  (
1  <  B  <->  1  < +oo ) )
6462, 63mpbiri 233 . . . . . . . 8  |-  ( B  = +oo  ->  1  <  B )
65 1z 10668 . . . . . . . . . 10  |-  1  e.  ZZ
66 zq 10951 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
6765, 66ax-mp 5 . . . . . . . . 9  |-  1  e.  QQ
68 breq2 4291 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( A  <  x  <->  A  <  1 ) )
69 breq1 4290 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  <  B  <->  1  <  B ) )
7068, 69anbi12d 710 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( A  <  x  /\  x  <  B )  <-> 
( A  <  1  /\  1  <  B ) ) )
7170rspcev 3068 . . . . . . . . 9  |-  ( ( 1  e.  QQ  /\  ( A  <  1  /\  1  <  B ) )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
7267, 71mpan 670 . . . . . . . 8  |-  ( ( A  <  1  /\  1  <  B )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
7360, 64, 72syl2an 477 . . . . . . 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 1159 . . . . . . . 8  |-  ( A  = -oo  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
7675, 1sylibr 212 . . . . . . 7  |-  ( A  = -oo  ->  A  e.  RR* )
7776, 29sylan 471 . . . . . 6  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
7855, 74, 773jaodan 1284 . . . . 5  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
792, 78sylan2b 475 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
8032, 38, 793jaoian 1283 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  ( A  < 
B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
811, 80sylanb 472 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
82813impia 1184 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 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2711   class class class wbr 4287  (class class class)co 6086   RRcr 9273   1c1 9275    + caddc 9277   +oocpnf 9407   -oocmnf 9408   RR*cxr 9409    < clt 9410    - cmin 9587   ZZcz 10638   QQcq 10945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946
This theorem is referenced by:  qextltlem  11164  xralrple  11167  ixxub  11313  ixxlb  11314  ioo0  11317  ico0  11338  ioc0  11339  blssps  19979  blss  19980  blcld  20060  qdensere  20329  tgqioo  20357  dvlip2  21447  lhop2  21467  itgsubst  21501  itg2gt0cn  28418
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