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Theorem qbtwnre 9213
Description: The rational numbers are dense in  RR: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.)
Assertion
Ref Expression
qbtwnre  |-  ( ( 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 qbtwnre
Dummy variables  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 916 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR )
2 simp1 915 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  RR )
31, 2resubcld 7451 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR )
4 simp3 917 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  <  B )
52, 1posdifd 7597 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  <  B  <->  0  <  ( B  -  A ) ) )
64, 5mpbid 139 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  0  <  ( B  -  A
) )
7 nnrecl 8237 . . 3  |-  ( ( ( B  -  A
)  e.  RR  /\  0  <  ( B  -  A ) )  ->  E. n  e.  NN  ( 1  /  n
)  <  ( B  -  A ) )
83, 6, 7syl2anc 397 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  E. n  e.  NN  ( 1  /  n )  <  ( B  -  A )
)
92adantr 265 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  A  e.  RR )
10 2re 8060 . . . . . . 7  |-  2  e.  RR
1110a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  2  e.  RR )
12 simprl 491 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  n  e.  NN )
1312nnred 8003 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  n  e.  RR )
1411, 13remulcld 7115 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  ( 2  x.  n )  e.  RR )
159, 14remulcld 7115 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  ( A  x.  ( 2  x.  n
) )  e.  RR )
16 rebtwn2z 9211 . . . 4  |-  ( ( A  x.  ( 2  x.  n ) )  e.  RR  ->  E. m  e.  ZZ  ( m  < 
( A  x.  (
2  x.  n ) )  /\  ( A  x.  ( 2  x.  n ) )  < 
( m  +  2 ) ) )
1715, 16syl 14 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  E. m  e.  ZZ  ( m  <  ( A  x.  ( 2  x.  n ) )  /\  ( A  x.  (
2  x.  n ) )  <  ( m  +  2 ) ) )
18 simprl 491 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  m  e.  ZZ )
19 2z 8330 . . . . . . 7  |-  2  e.  ZZ
2019a1i 9 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  2  e.  ZZ )
2118, 20zaddcld 8423 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  ( m  +  2 )  e.  ZZ )
22 2nn 8144 . . . . . . 7  |-  2  e.  NN
2322a1i 9 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  2  e.  NN )
2412adantr 265 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  n  e.  NN )
2523, 24nnmulcld 8038 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  ( 2  x.  n )  e.  NN )
26 znq 8656 . . . . 5  |-  ( ( ( m  +  2 )  e.  ZZ  /\  ( 2  x.  n
)  e.  NN )  ->  ( ( m  +  2 )  / 
( 2  x.  n
) )  e.  QQ )
2721, 25, 26syl2anc 397 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  ( (
m  +  2 )  /  ( 2  x.  n ) )  e.  QQ )
28 simprrr 500 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  ( A  x.  ( 2  x.  n
) )  <  (
m  +  2 ) )
299adantr 265 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  A  e.  RR )
3021zred 8419 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  ( m  +  2 )  e.  RR )
3125nnrpd 8719 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  ( 2  x.  n )  e.  RR+ )
3229, 30, 31ltmuldivd 8768 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  ( ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 )  <->  A  <  ( ( m  +  2 )  /  ( 2  x.  n ) ) ) )
3328, 32mpbid 139 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  A  <  ( ( m  +  2 )  /  ( 2  x.  n ) ) )
34 simpll2 955 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  B  e.  RR )
35 simprrl 499 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  m  <  ( A  x.  ( 2  x.  n ) ) )
36 simplrr 496 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  ( 1  /  n )  < 
( B  -  A
) )
3718, 24, 29, 34, 35, 36qbtwnrelemcalc 9212 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  ( (
m  +  2 )  /  ( 2  x.  n ) )  < 
B )
38 breq2 3796 . . . . . 6  |-  ( x  =  ( ( m  +  2 )  / 
( 2  x.  n
) )  ->  ( A  <  x  <->  A  <  ( ( m  +  2 )  /  ( 2  x.  n ) ) ) )
39 breq1 3795 . . . . . 6  |-  ( x  =  ( ( m  +  2 )  / 
( 2  x.  n
) )  ->  (
x  <  B  <->  ( (
m  +  2 )  /  ( 2  x.  n ) )  < 
B ) )
4038, 39anbi12d 450 . . . . 5  |-  ( x  =  ( ( m  +  2 )  / 
( 2  x.  n
) )  ->  (
( A  <  x  /\  x  <  B )  <-> 
( A  <  (
( m  +  2 )  /  ( 2  x.  n ) )  /\  ( ( m  +  2 )  / 
( 2  x.  n
) )  <  B
) ) )
4140rspcev 2673 . . . 4  |-  ( ( ( ( m  + 
2 )  /  (
2  x.  n ) )  e.  QQ  /\  ( A  <  ( ( m  +  2 )  /  ( 2  x.  n ) )  /\  ( ( m  + 
2 )  /  (
2  x.  n ) )  <  B ) )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
4227, 33, 37, 41syl12anc 1144 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  < 
B )  /\  (
n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  /\  ( m  e.  ZZ  /\  ( m  <  ( A  x.  ( 2  x.  n
) )  /\  ( A  x.  ( 2  x.  n ) )  <  ( m  + 
2 ) ) ) )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  B ) )
4317, 42rexlimddv 2454 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  B ) )
448, 43rexlimddv 2454 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:    -> wi 4    /\ wa 101    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324   class class class wbr 3792  (class class class)co 5540   RRcr 6946   0cc0 6947   1c1 6948    + caddc 6950    x. cmul 6952    < clt 7119    - cmin 7245    / cdiv 7725   NNcn 7990   2c2 8040   ZZcz 8302   QQcq 8651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060  ax-arch 7061
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-inn 7991  df-2 8049  df-n0 8240  df-z 8303  df-uz 8570  df-q 8652  df-rp 8682
This theorem is referenced by:  qbtwnxr  9214  qdenre  10029
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