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Theorem qbtwnxr 11158
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 11084 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 elxr 11084 . . . . 5  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3 qbtwnre 11157 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
433expia 1182 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
5 simpl 454 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  e.  RR )
6 peano2re 9530 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
76adantr 462 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  ( A  +  1 )  e.  RR )
8 ltp1 10155 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  <  ( A  +  1 ) )
98adantr 462 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  A  <  ( A  +  1 ) )
10 qbtwnre 11157 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( A  +  1
)  e.  RR  /\  A  <  ( A  + 
1 ) )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
115, 7, 9, 10syl3anc 1211 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  = +oo )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  ( A  + 
1 ) ) )
12 qre 10946 . . . . . . . . . . . . . 14  |-  ( x  e.  QQ  ->  x  e.  RR )
13 ltpnf 11090 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  x  < +oo )
1412, 13syl 16 . . . . . . . . . . . . 13  |-  ( x  e.  QQ  ->  x  < +oo )
1514adantl 463 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  x  < +oo )
16 simplr 747 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  B  = +oo )
1715, 16breqtrrd 4306 . . . . . . . . . . 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 560 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  = +oo )  /\  x  e.  QQ )  ->  ( ( A  <  x  /\  x  <  ( A  +  1 ) )  ->  ( A  <  x  /\  x  <  B ) ) )
2019reximdva 2818 . . . . . . . 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 9417 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
24 breq2 4284 . . . . . . . . 9  |-  ( B  = -oo  ->  ( A  <  B  <->  A  < -oo ) )
2524adantl 463 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  = -oo )  ->  ( A  <  B  <->  A  < -oo ) )
26 nltmnf 11097 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  -.  A  < -oo )
2726adantr 462 . . . . . . . . 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 468 . . . . . 6  |-  ( ( A  e.  RR  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
314, 22, 303jaodan 1277 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
322, 31sylan2b 472 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
33 breq1 4283 . . . . . 6  |-  ( A  = +oo  ->  ( A  <  B  <-> +oo  <  B
) )
3433adantr 462 . . . . 5  |-  ( ( A  = +oo  /\  B  e.  RR* )  -> 
( A  <  B  <-> +oo 
<  B ) )
35 pnfnlt 11096 . . . . . . 7  |-  ( B  e.  RR*  ->  -. +oo  <  B )
3635adantl 463 . . . . . 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 9663 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
4039adantl 463 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  -  1 )  e.  RR )
41 simpr 458 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  B  e.  RR )
42 ltm1 10157 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  -  1 )  <  B )
4342adantl 463 . . . . . . . . 9  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  ( B  -  1 )  <  B )
44 qbtwnre 11157 . . . . . . . . 9  |-  ( ( ( B  -  1 )  e.  RR  /\  B  e.  RR  /\  ( B  -  1 )  <  B )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
4540, 41, 43, 44syl3anc 1211 . . . . . . . 8  |-  ( ( A  = -oo  /\  B  e.  RR )  ->  E. x  e.  QQ  ( ( B  - 
1 )  <  x  /\  x  <  B ) )
46 simpll 746 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  A  = -oo )
4712adantl 463 . . . . . . . . . . . . 13  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  x  e.  RR )
48 mnflt 11092 . . . . . . . . . . . . 13  |-  ( x  e.  RR  -> -oo  <  x )
4947, 48syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  -> -oo  <  x )
5046, 49eqbrtrd 4300 . . . . . . . . . . 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 559 . . . . . . . . 9  |-  ( ( ( A  = -oo  /\  B  e.  RR )  /\  x  e.  QQ )  ->  ( ( ( B  -  1 )  <  x  /\  x  <  B )  ->  ( A  <  x  /\  x  <  B ) ) )
5352reximdva 2818 . . . . . . . 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 9373 . . . . . . . . . 10  |-  1  e.  RR
57 mnflt 11092 . . . . . . . . . 10  |-  ( 1  e.  RR  -> -oo  <  1 )
5856, 57ax-mp 5 . . . . . . . . 9  |- -oo  <  1
59 breq1 4283 . . . . . . . . 9  |-  ( A  = -oo  ->  ( A  <  1  <-> -oo  <  1
) )
6058, 59mpbiri 233 . . . . . . . 8  |-  ( A  = -oo  ->  A  <  1 )
61 ltpnf 11090 . . . . . . . . . 10  |-  ( 1  e.  RR  ->  1  < +oo )
6256, 61ax-mp 5 . . . . . . . . 9  |-  1  < +oo
63 breq2 4284 . . . . . . . . 9  |-  ( B  = +oo  ->  (
1  <  B  <->  1  < +oo ) )
6462, 63mpbiri 233 . . . . . . . 8  |-  ( B  = +oo  ->  1  <  B )
65 1z 10664 . . . . . . . . . 10  |-  1  e.  ZZ
66 zq 10947 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
6765, 66ax-mp 5 . . . . . . . . 9  |-  1  e.  QQ
68 breq2 4284 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( A  <  x  <->  A  <  1 ) )
69 breq1 4283 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
x  <  B  <->  1  <  B ) )
7068, 69anbi12d 703 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( A  <  x  /\  x  <  B )  <-> 
( A  <  1  /\  1  <  B ) ) )
7170rspcev 3062 . . . . . . . . 9  |-  ( ( 1  e.  QQ  /\  ( A  <  1  /\  1  <  B ) )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
7267, 71mpan 663 . . . . . . . 8  |-  ( ( A  <  1  /\  1  <  B )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
7360, 64, 72syl2an 474 . . . . . . 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 1152 . . . . . . . 8  |-  ( A  = -oo  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
7675, 1sylibr 212 . . . . . . 7  |-  ( A  = -oo  ->  A  e.  RR* )
7776, 29sylan 468 . . . . . 6  |-  ( ( A  = -oo  /\  B  = -oo )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
7855, 74, 773jaodan 1277 . . . . 5  |-  ( ( A  = -oo  /\  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
792, 78sylan2b 472 . . . 4  |-  ( ( A  = -oo  /\  B  e.  RR* )  -> 
( A  <  B  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) ) )
8032, 38, 793jaoian 1276 . . 3  |-  ( ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  /\  B  e.  RR* )  ->  ( A  < 
B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
811, 80sylanb 469 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) ) )
82813impia 1177 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 957    /\ w3a 958    = wceq 1362    e. wcel 1755   E.wrex 2706   class class class wbr 4280  (class class class)co 6080   RRcr 9269   1c1 9271    + caddc 9273   +oocpnf 9403   -oocmnf 9404   RR*cxr 9405    < clt 9406    - cmin 9583   ZZcz 10634   QQcq 10941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-q 10942
This theorem is referenced by:  qextltlem  11160  xralrple  11163  ixxub  11309  ixxlb  11310  ioo0  11313  ico0  11334  ioc0  11335  blssps  19841  blss  19842  blcld  19922  qdensere  20191  tgqioo  20219  dvlip2  21309  lhop2  21329  itgsubst  21363  itg2gt0cn  28291
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