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Theorem qbtwnre 10227
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 8312 . . 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 8463 . . . 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 9147 . . 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 8962 . . . . . . 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 8905 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  n  e.  RR )
1411, 13remulcld 7962 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( n  e.  NN  /\  ( 1  /  n
)  <  ( B  -  A ) ) )  ->  ( 2  x.  n )  e.  RR )
159, 14remulcld 7962 . . . 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 10225 . . . 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 9254 . . . . . . 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 9352 . . . . 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 9053 . . . . . . 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 8941 . . . . 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 9597 . . . . 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 9348 . . . . . 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 9665 . . . . . 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 9715 . . . . 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 10226 . . . 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 4002 . . . . . 6  |-  ( x  =  ( ( m  +  2 )  / 
( 2  x.  n
) )  ->  ( A  <  x  <->  A  <  ( ( m  +  2 )  /  ( 2  x.  n ) ) ) )
39 breq1 4001 . . . . . 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 2839 . . . 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 2597 . 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 2597 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 2146   E.wrex 2454   class class class wbr 3998  (class class class)co 5865   RRcr 7785   0cc0 7786   1c1 7787    + caddc 7789    x. cmul 7791    < clt 7966    - cmin 8102    / cdiv 8602   NNcn 8892   2c2 8943   ZZcz 9226   QQcq 9592
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-n0 9150  df-z 9227  df-uz 9502  df-q 9593  df-rp 9625
This theorem is referenced by:  qbtwnxr  10228  qdenre  11179  expcnvre  11479
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