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Theorem qbtwnxr 10543
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 10474 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
2 elxr 10474 . . . . 5  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = 
+oo  \/  B  =  -oo ) )
3 qbtwnre 10542 . . . . . . 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 9001 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
76adantr 451 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( A  +  1 )  e.  RR )
8 ltp1 9610 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
98adantr 451 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  A  <  ( A  + 
1 ) )
10 qbtwnre 10542 . . . . . . . . 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 10337 . . . . . . . . . . . . . 14  |-  ( x  e.  QQ  ->  x  e.  RR )
13 ltpnf 10479 . . . . . . . . . . . . . 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 4065 . . . . . . . . . . 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 2668 . . . . . . . 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 8893 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
24 breq2 4043 . . . . . . . . 9  |-  ( B  =  -oo  ->  ( A  <  B  <->  A  <  -oo ) )
2524adantl 452 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  =  -oo )  ->  ( A  <  B  <->  A  <  -oo ) )
26 nltmnf 10484 . . . . . . . . . 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 4042 . . . . . 6  |-  ( A  =  +oo  ->  ( A  <  B  <->  +oo  <  B
) )
3433adantr 451 . . . . 5  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
( A  <  B  <->  +oo 
<  B ) )
35 pnfnlt 10483 . . . . . . 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 9129 . . . . . . . . . 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 9612 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  <  B )
4342adantl 452 . . . . . . . . 9  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( B  -  1 )  <  B )
44 qbtwnre 10542 . . . . . . . . 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 10480 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  -oo  <  x )
4947, 48syl 15 . . . . . . . . . . . 12  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  -oo  <  x )
5046, 49eqbrtrd 4059 . . . . . . . . . . 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 2668 . . . . . . . 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 8853 . . . . . . . . . 10  |-  1  e.  RR
57 mnflt 10480 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  -oo  <  1 )
5856, 57ax-mp 8 . . . . . . . . 9  |-  -oo  <  1
59 breq1 4042 . . . . . . . . 9  |-  ( A  =  -oo  ->  ( A  <  1  <->  -oo  <  1
) )
6058, 59mpbiri 224 . . . . . . . 8  |-  ( A  =  -oo  ->  A  <  1 )
61 ltpnf 10479 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  1  <  +oo )
6256, 61ax-mp 8 . . . . . . . . 9  |-  1  <  +oo
63 breq2 4043 . . . . . . . . 9  |-  ( B  =  +oo  ->  (
1  <  B  <->  1  <  +oo ) )
6462, 63mpbiri 224 . . . . . . . 8  |-  ( B  =  +oo  ->  1  <  B )
65 1z 10069 . . . . . . . . . 10  |-  1  e.  ZZ
66 zq 10338 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
6765, 66ax-mp 8 . . . . . . . . 9  |-  1  e.  QQ
68 breq2 4043 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( A  <  x  <->  A  <  1 ) )
69 breq1 4042 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  <  B  <->  1  <  B ) )
7068, 69anbi12d 691 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( A  <  x  /\  x  <  B )  <-> 
( A  <  1  /\  1  <  B ) ) )
7170rspcev 2897 . . . . . . . . 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 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039  (class class class)co 5874   RRcr 8752   1c1 8754    + caddc 8756    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883    - cmin 9053   ZZcz 10040   QQcq 10332
This theorem is referenced by:  qextltlem  10545  xralrple  10548  ixxub  10693  ixxlb  10694  ioo0  10697  ico0  10718  ioc0  10719  blss  17988  blcld  18067  qdensere  18295  tgqioo  18322  dvlip2  19358  lhop2  19378  itgsubst  19412  itg2gt0cn  25006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333
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