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Theorem qbtwnre 10243
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 998 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR )
2 simp1 997 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  RR )
31, 2resubcld 8328 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR )
4 simp3 999 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  <  B )
52, 1posdifd 8479 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A  <  B  <->  0  <  ( B  -  A ) ) )
64, 5mpbid 147 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  0  <  ( B  -  A
) )
7 nnrecl 9163 . . 3  |-  ( ( ( B  -  A
)  e.  RR  /\  0  <  ( B  -  A ) )  ->  E. n  e.  NN  ( 1  /  n
)  <  ( B  -  A ) )
83, 6, 7syl2anc 411 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  E. n  e.  NN  ( 1  /  n )  <  ( B  -  A )
)
92adantr 276 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  A  e.  RR )
10 2re 8978 . . . . . . 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 529 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  n  e.  NN )
1312nnred 8921 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  n  e.  RR )
1411, 13remulcld 7978 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  ( 2  x.  n )  e.  RR )
159, 14remulcld 7978 . . . 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 10241 . . . 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 529 . . . . . 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 9270 . . . . . . 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 9368 . . . . 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 9069 . . . . . . 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 276 . . . . . 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 8957 . . . . 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 9613 . . . . 5  |-  ( ( ( m  +  2 )  e.  ZZ  /\  ( 2  x.  n
)  e.  NN )  ->  ( ( m  +  2 )  / 
( 2  x.  n
) )  e.  QQ )
2721, 25, 26syl2anc 411 . . . 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 540 . . . . 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 276 . . . . . 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 9364 . . . . . 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 9681 . . . . . 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 9731 . . . . 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 147 . . . 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 1037 . . . . 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 539 . . . . 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 536 . . . . 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 10242 . . . 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 4004 . . . . . 6  |-  ( x  =  ( ( m  +  2 )  / 
( 2  x.  n
) )  ->  ( A  <  x  <->  A  <  ( ( m  +  2 )  /  ( 2  x.  n ) ) ) )
39 breq1 4003 . . . . . 6  |-  ( x  =  ( ( m  +  2 )  / 
( 2  x.  n
) )  ->  (
x  <  B  <->  ( (
m  +  2 )  /  ( 2  x.  n ) )  < 
B ) )
4038, 39anbi12d 473 . . . . 5  |-  ( x  =  ( ( m  +  2 )  / 
( 2  x.  n
) )  ->  (
( A  <  x  /\  x  <  B )  <-> 
( A  <  (
( m  +  2 )  /  ( 2  x.  n ) )  /\  ( ( m  +  2 )  / 
( 2  x.  n
) )  <  B
) ) )
4140rspcev 2841 . . . 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 1236 . . 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 2599 . 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 2599 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 104    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456   class class class wbr 4000  (class class class)co 5869   RRcr 7801   0cc0 7802   1c1 7803    + caddc 7805    x. cmul 7807    < clt 7982    - cmin 8118    / cdiv 8618   NNcn 8908   2c2 8959   ZZcz 9242   QQcq 9608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641
This theorem is referenced by:  qbtwnxr  10244  qdenre  11195  expcnvre  11495
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