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Theorem qbtwnxr 11490
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 11413 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11413 . . . . 5  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 qbtwnre 11489 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
433expia 1209 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
5 simpl 459 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  e.  RR )
6 peano2re 9803 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
76adantr 467 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  +  1 )  e.  RR )
8 ltp1 10440 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
98adantr 467 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  <  ( A  +  1 ) )
10 qbtwnre 11489 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( A  +  1
)  e.  RR  /\  A  <  ( A  + 
1 ) )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
115, 7, 9, 10syl3anc 1267 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
12 qre 11266 . . . . . . . . . . . . . 14  |-  ( x  e.  QQ  ->  x  e.  RR )
13 ltpnf 11419 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  x  < +oo )
1412, 13syl 17 . . . . . . . . . . . . 13  |-  ( x  e.  QQ  ->  x  < +oo )
1514adantl 468 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  x  < +oo )
16 simplr 761 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  B  = +oo )
1715, 16breqtrrd 4428 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  x  <  B
)
1817a1d 26 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  ( x  < 
( A  +  1 )  ->  x  <  B ) )
1918anim2d 568 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  ( ( A  <  x  /\  x  <  ( A  +  1 ) )  ->  ( A  <  x  /\  x  <  B ) ) )
2019reximdva 2861 . . . . . . . 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 26 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
23 rexr 9683 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
24 breq2 4405 . . . . . . . . 9  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
2524adantl 468 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
26 nltmnf 11428 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  -.  A  < -oo )
2726adantr 467 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  -.  A  < -oo )
2827pm2.21d 110 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  ( A  < -oo  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
2925, 28sylbid 219 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
3023, 29sylan 474 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
314, 22, 303jaodan 1333 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
322, 31sylan2b 478 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
33 breq1 4404 . . . . . 6  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
3433adantr 467 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
35 pnfnlt 11427 . . . . . . 7  |-  ( B  e.  RR*  ->  -. +oo  <  B )
3635adantl 468 . . . . . 6  |-  ( ( A  = +oo  /\  B  e.  RR* )  ->  -. +oo  <  B )
3736pm2.21d 110 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( +oo  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
3834, 37sylbid 219 . . . 4  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
39 peano2rem 9938 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
4039adantl 468 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  -  1 )  e.  RR )
41 simpr 463 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  B  e.  RR )
42 ltm1 10442 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  <  B )
4342adantl 468 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  -  1 )  <  B )
44 qbtwnre 11489 . . . . . . . . 9  |-  ( ( ( B  -  1 )  e.  RR  /\  B  e.  RR  /\  ( B  -  1 )  <  B )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
4540, 41, 43, 44syl3anc 1267 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
46 simpll 759 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  A  = -oo )
4712adantl 468 . . . . . . . . . . . . 13  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  x  e.  RR )
48 mnflt 11422 . . . . . . . . . . . . 13  |-  ( x  e.  RR  -> -oo  <  x )
4947, 48syl 17 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  -> -oo  <  x )
5046, 49eqbrtrd 4422 . . . . . . . . . . 11  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  A  <  x
)
5150a1d 26 . . . . . . . . . 10  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  ( ( B  -  1 )  < 
x  ->  A  <  x ) )
5251anim1d 567 . . . . . . . . 9  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  ( ( ( B  -  1 )  <  x  /\  x  <  B )  ->  ( A  <  x  /\  x  <  B ) ) )
5352reximdva 2861 . . . . . . . 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 26 . . . . . 6  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
56 1re 9639 . . . . . . . . . 10  |-  1  e.  RR
57 mnflt 11422 . . . . . . . . . 10  |-  ( 1  e.  RR  -> -oo  <  1 )
5856, 57ax-mp 5 . . . . . . . . 9  |- -oo  <  1
59 breq1 4404 . . . . . . . . 9  |-  ( A  = -oo  ->  ( A  <  1  <-> -oo  <  1
) )
6058, 59mpbiri 237 . . . . . . . 8  |-  ( A  = -oo  ->  A  <  1 )
61 ltpnf 11419 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  1  < +oo )
6256, 61ax-mp 5 . . . . . . . . 9  |-  1  < +oo
63 breq2 4405 . . . . . . . . 9  |-  ( B  = +oo  ->  (
1  <  B  <->  1  < +oo ) )
6462, 63mpbiri 237 . . . . . . . 8  |-  ( B  = +oo  ->  1  <  B )
65 1z 10964 . . . . . . . . . 10  |-  1  e.  ZZ
66 zq 11267 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
6765, 66ax-mp 5 . . . . . . . . 9  |-  1  e.  QQ
68 breq2 4405 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( A  <  x  <->  A  <  1 ) )
69 breq1 4404 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  <  B  <->  1  <  B ) )
7068, 69anbi12d 716 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( A  <  x  /\  x  <  B )  <-> 
( A  <  1  /\  1  <  B ) ) )
7170rspcev 3149 . . . . . . . . 9  |-  ( ( 1  e.  QQ  /\  ( A  <  1  /\  1  <  B ) )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
7267, 71mpan 675 . . . . . . . 8  |-  ( ( A  <  1  /\  1  <  B )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
7360, 64, 72syl2an 480 . . . . . . 7  |-  ( ( A  = -oo  /\  B  = +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
7473a1d 26 . . . . . 6  |-  ( ( A  = -oo  /\  B  = +oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
75 3mix3 1178 . . . . . . . 8  |-  ( A  = -oo  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
7675, 1sylibr 216 . . . . . . 7  |-  ( A  = -oo  ->  A  e.  RR* )
7776, 29sylan 474 . . . . . 6  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
7855, 74, 773jaodan 1333 . . . . 5  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
792, 78sylan2b 478 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
8032, 38, 793jaoian 1332 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  ( A  < 
B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
811, 80sylanb 475 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
82813impia 1204 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 188    /\ wa 371    \/ w3o 983    /\ w3a 984    = wceq 1443    e. wcel 1886   E.wrex 2737   class class class wbr 4401  (class class class)co 6288   RRcr 9535   1c1 9537    + caddc 9539   +oocpnf 9669   -oocmnf 9670   RR*cxr 9671    < clt 9672    - cmin 9857   ZZcz 10934   QQcq 11261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262
This theorem is referenced by:  qextltlem  11492  xralrple  11495  ixxub  11653  ixxlb  11654  ixxlbOLD  11655  ioo0  11658  ico0  11679  ioc0  11680  blssps  21432  blss  21433  blcld  21513  qdensere  21783  tgqioo  21811  dvlip2  22940  lhop2  22960  itgsubst  22994  itg2gt0cn  31990  qinioo  37631  qndenserrnbllem  38157
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