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Theorem qbtwnxr 10788
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 10718 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
2 elxr 10718 . . . . 5  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = 
+oo  \/  B  =  -oo ) )
3 qbtwnre 10787 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
433expia 1156 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
5 simpl 445 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  A  e.  RR )
6 peano2re 9241 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
76adantr 453 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( A  +  1 )  e.  RR )
8 ltp1 9850 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
98adantr 453 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  A  <  ( A  + 
1 ) )
10 qbtwnre 10787 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( A  +  1
)  e.  RR  /\  A  <  ( A  + 
1 ) )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
115, 7, 9, 10syl3anc 1185 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  =  +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
12 qre 10581 . . . . . . . . . . . . . 14  |-  ( x  e.  QQ  ->  x  e.  RR )
13 ltpnf 10723 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  x  <  +oo )
1412, 13syl 16 . . . . . . . . . . . . 13  |-  ( x  e.  QQ  ->  x  <  +oo )
1514adantl 454 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  =  +oo )  /\  x  e.  QQ )  ->  x  <  +oo )
16 simplr 733 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  =  +oo )  /\  x  e.  QQ )  ->  B  =  +oo )
1715, 16breqtrrd 4240 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  =  +oo )  /\  x  e.  QQ )  ->  x  <  B
)
1817a1d 24 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  =  +oo )  /\  x  e.  QQ )  ->  ( x  < 
( A  +  1 )  ->  x  <  B ) )
1918anim2d 550 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  =  +oo )  /\  x  e.  QQ )  ->  ( ( A  <  x  /\  x  <  ( A  +  1 ) )  ->  ( A  <  x  /\  x  <  B ) ) )
2019reximdva 2820 . . . . . . . 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 24 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  +oo )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
23 rexr 9132 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
24 breq2 4218 . . . . . . . . 9  |-  ( B  =  -oo  ->  ( A  <  B  <->  A  <  -oo ) )
2524adantl 454 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  =  -oo )  ->  ( A  <  B  <->  A  <  -oo ) )
26 nltmnf 10728 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  -.  A  <  -oo )
2726adantr 453 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  =  -oo )  ->  -.  A  <  -oo )
2827pm2.21d 101 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  =  -oo )  ->  ( A  <  -oo  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
2925, 28sylbid 208 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  =  -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
3023, 29sylan 459 . . . . . 6  |-  ( ( A  e.  RR  /\  B  =  -oo )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
314, 22, 303jaodan 1251 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
322, 31sylan2b 463 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
33 breq1 4217 . . . . . 6  |-  ( A  =  +oo  ->  ( A  <  B  <->  +oo  <  B
) )
3433adantr 453 . . . . 5  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
( A  <  B  <->  +oo 
<  B ) )
35 pnfnlt 10727 . . . . . . 7  |-  ( B  e.  RR*  ->  -.  +oo  <  B )
3635adantl 454 . . . . . 6  |-  ( ( A  =  +oo  /\  B  e.  RR* )  ->  -.  +oo  <  B )
3736pm2.21d 101 . . . . 5  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
(  +oo  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
3834, 37sylbid 208 . . . 4  |-  ( ( A  =  +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
39 peano2rem 9369 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
4039adantl 454 . . . . . . . . 9  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( B  -  1 )  e.  RR )
41 simpr 449 . . . . . . . . 9  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  B  e.  RR )
42 ltm1 9852 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  <  B )
4342adantl 454 . . . . . . . . 9  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( B  -  1 )  <  B )
44 qbtwnre 10787 . . . . . . . . 9  |-  ( ( ( B  -  1 )  e.  RR  /\  B  e.  RR  /\  ( B  -  1 )  <  B )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
4540, 41, 43, 44syl3anc 1185 . . . . . . . 8  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
46 simpll 732 . . . . . . . . . . . 12  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  A  =  -oo )
4712adantl 454 . . . . . . . . . . . . 13  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  x  e.  RR )
48 mnflt 10724 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  -oo  <  x )
4947, 48syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  -oo  <  x )
5046, 49eqbrtrd 4234 . . . . . . . . . . 11  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  A  <  x
)
5150a1d 24 . . . . . . . . . 10  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  ( ( B  -  1 )  < 
x  ->  A  <  x ) )
5251anim1d 549 . . . . . . . . 9  |-  ( ( ( A  =  -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  ( ( ( B  -  1 )  <  x  /\  x  <  B )  ->  ( A  <  x  /\  x  <  B ) ) )
5352reximdva 2820 . . . . . . . 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 24 . . . . . 6  |-  ( ( A  =  -oo  /\  B  e.  RR )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
56 1re 9092 . . . . . . . . . 10  |-  1  e.  RR
57 mnflt 10724 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  -oo  <  1 )
5856, 57ax-mp 8 . . . . . . . . 9  |-  -oo  <  1
59 breq1 4217 . . . . . . . . 9  |-  ( A  =  -oo  ->  ( A  <  1  <->  -oo  <  1
) )
6058, 59mpbiri 226 . . . . . . . 8  |-  ( A  =  -oo  ->  A  <  1 )
61 ltpnf 10723 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  1  <  +oo )
6256, 61ax-mp 8 . . . . . . . . 9  |-  1  <  +oo
63 breq2 4218 . . . . . . . . 9  |-  ( B  =  +oo  ->  (
1  <  B  <->  1  <  +oo ) )
6462, 63mpbiri 226 . . . . . . . 8  |-  ( B  =  +oo  ->  1  <  B )
65 1z 10313 . . . . . . . . . 10  |-  1  e.  ZZ
66 zq 10582 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
6765, 66ax-mp 8 . . . . . . . . 9  |-  1  e.  QQ
68 breq2 4218 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( A  <  x  <->  A  <  1 ) )
69 breq1 4217 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  <  B  <->  1  <  B ) )
7068, 69anbi12d 693 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( A  <  x  /\  x  <  B )  <-> 
( A  <  1  /\  1  <  B ) ) )
7170rspcev 3054 . . . . . . . . 9  |-  ( ( 1  e.  QQ  /\  ( A  <  1  /\  1  <  B ) )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
7267, 71mpan 653 . . . . . . . 8  |-  ( ( A  <  1  /\  1  <  B )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
7360, 64, 72syl2an 465 . . . . . . 7  |-  ( ( A  =  -oo  /\  B  =  +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
7473a1d 24 . . . . . 6  |-  ( ( A  =  -oo  /\  B  =  +oo )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
75 3mix3 1129 . . . . . . . 8  |-  ( A  =  -oo  ->  ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo ) )
7675, 1sylibr 205 . . . . . . 7  |-  ( A  =  -oo  ->  A  e.  RR* )
7776, 29sylan 459 . . . . . 6  |-  ( ( A  =  -oo  /\  B  =  -oo )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
7855, 74, 773jaodan 1251 . . . . 5  |-  ( ( A  =  -oo  /\  ( B  e.  RR  \/  B  =  +oo  \/  B  =  -oo )
)  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
792, 78sylan2b 463 . . . 4  |-  ( ( A  =  -oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
8032, 38, 793jaoian 1250 . . 3  |-  ( ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
811, 80sylanb 460 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
82813impia 1151 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 178    /\ wa 360    \/ w3o 936    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4214  (class class class)co 6083   RRcr 8991   1c1 8993    + caddc 8995    +oocpnf 9119    -oocmnf 9120   RR*cxr 9121    < clt 9122    - cmin 9293   ZZcz 10284   QQcq 10576
This theorem is referenced by:  qextltlem  10790  xralrple  10793  ixxub  10939  ixxlb  10940  ioo0  10943  ico0  10964  ioc0  10965  blssps  18456  blss  18457  blcld  18537  qdensere  18806  tgqioo  18833  dvlip2  19881  lhop2  19901  itgsubst  19935  itg2gt0cn  26262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577
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