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Theorem qbtwnxr 10065
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 9592 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 9592 . . . . 5  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 qbtwnre 10064 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
433expia 1184 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
5 simpl 108 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  e.  RR )
6 peano2re 7921 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
76adantr 274 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  +  1 )  e.  RR )
8 ltp1 8625 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
98adantr 274 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  <  ( A  +  1 ) )
10 qbtwnre 10064 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( A  +  1
)  e.  RR  /\  A  <  ( A  + 
1 ) )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
115, 7, 9, 10syl3anc 1217 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
12 qre 9443 . . . . . . . . . . . . . 14  |-  ( x  e.  QQ  ->  x  e.  RR )
13 ltpnf 9596 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  x  < +oo )
1412, 13syl 14 . . . . . . . . . . . . 13  |-  ( x  e.  QQ  ->  x  < +oo )
1514adantl 275 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  x  < +oo )
16 simplr 520 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  B  = +oo )
1715, 16breqtrrd 3963 . . . . . . . . . . 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 335 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  ( ( A  <  x  /\  x  <  ( A  +  1 ) )  ->  ( A  <  x  /\  x  <  B ) ) )
2019reximdva 2537 . . . . . . . 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 13 . . . . . . 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 7834 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
24 breq2 3940 . . . . . . . . 9  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
2524adantl 275 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
26 nltmnf 9603 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  -.  A  < -oo )
2726adantr 274 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  -.  A  < -oo )
2827pm2.21d 609 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  ( A  < -oo  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
2925, 28sylbid 149 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
3023, 29sylan 281 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
314, 22, 303jaodan 1285 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
322, 31sylan2b 285 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
33 breq1 3939 . . . . . 6  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
3433adantr 274 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
35 pnfnlt 9602 . . . . . . 7  |-  ( B  e.  RR*  ->  -. +oo  <  B )
3635adantl 275 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
3736pm2.21d 609 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( +oo  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
3834, 37sylbid 149 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
39 peano2rem 8052 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
4039adantl 275 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  -  1 )  e.  RR )
41 simpr 109 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  B  e.  RR )
42 ltm1 8627 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  <  B )
4342adantl 275 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  -  1 )  <  B )
44 qbtwnre 10064 . . . . . . . . 9  |-  ( ( ( B  -  1 )  e.  RR  /\  B  e.  RR  /\  ( B  -  1 )  <  B )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
4540, 41, 43, 44syl3anc 1217 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
46 simpll 519 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  A  = -oo )
4712adantl 275 . . . . . . . . . . . . 13  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  x  e.  RR )
48 mnflt 9598 . . . . . . . . . . . . 13  |-  ( x  e.  RR  -> -oo  <  x )
4947, 48syl 14 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  -> -oo  <  x )
5046, 49eqbrtrd 3957 . . . . . . . . . . 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 334 . . . . . . . . 9  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  ( ( ( B  -  1 )  <  x  /\  x  <  B )  ->  ( A  <  x  /\  x  <  B ) ) )
5352reximdva 2537 . . . . . . . 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 13 . . . . . . 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 7788 . . . . . . . . . 10  |-  1  e.  RR
57 mnflt 9598 . . . . . . . . . 10  |-  ( 1  e.  RR  -> -oo  <  1 )
5856, 57ax-mp 5 . . . . . . . . 9  |- -oo  <  1
59 breq1 3939 . . . . . . . . 9  |-  ( A  = -oo  ->  ( A  <  1  <-> -oo  <  1
) )
6058, 59mpbiri 167 . . . . . . . 8  |-  ( A  = -oo  ->  A  <  1 )
61 ltpnf 9596 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  1  < +oo )
6256, 61ax-mp 5 . . . . . . . . 9  |-  1  < +oo
63 breq2 3940 . . . . . . . . 9  |-  ( B  = +oo  ->  (
1  <  B  <->  1  < +oo ) )
6462, 63mpbiri 167 . . . . . . . 8  |-  ( B  = +oo  ->  1  <  B )
65 1z 9103 . . . . . . . . . 10  |-  1  e.  ZZ
66 zq 9444 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
6765, 66ax-mp 5 . . . . . . . . 9  |-  1  e.  QQ
68 breq2 3940 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( A  <  x  <->  A  <  1 ) )
69 breq1 3939 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  <  B  <->  1  <  B ) )
7068, 69anbi12d 465 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( A  <  x  /\  x  <  B )  <-> 
( A  <  1  /\  1  <  B ) ) )
7170rspcev 2792 . . . . . . . . 9  |-  ( ( 1  e.  QQ  /\  ( A  <  1  /\  1  <  B ) )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
7267, 71mpan 421 . . . . . . . 8  |-  ( ( A  <  1  /\  1  <  B )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
7360, 64, 72syl2an 287 . . . . . . 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 1153 . . . . . . . 8  |-  ( A  = -oo  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
7675, 1sylibr 133 . . . . . . 7  |-  ( A  = -oo  ->  A  e.  RR* )
7776, 29sylan 281 . . . . . 6  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
7855, 74, 773jaodan 1285 . . . . 5  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
792, 78sylan2b 285 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
8032, 38, 793jaoian 1284 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  ( A  < 
B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
811, 80sylanb 282 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
82813impia 1179 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    /\ wa 103    <-> wb 104    \/ w3o 962    /\ w3a 963    = wceq 1332    e. wcel 1481   E.wrex 2418   class class class wbr 3936  (class class class)co 5781   RRcr 7642   1c1 7644    + caddc 7646   +oocpnf 7820   -oocmnf 7821   RR*cxr 7822    < clt 7823    - cmin 7956   ZZcz 9077   QQcq 9437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-po 4225  df-iso 4226  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-n0 9001  df-z 9078  df-uz 9350  df-q 9438  df-rp 9470
This theorem is referenced by:  ioo0  10067  ioom  10068  ico0  10069  ioc0  10070  blssps  12633  blss  12634  tgqioo  12753
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