ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltbtwnnqq Unicode version

Theorem ltbtwnnqq 6667
Description: There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by Jim Kingdon, 24-Sep-2019.)
Assertion
Ref Expression
ltbtwnnqq  |-  ( A 
<Q  B  <->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem ltbtwnnqq
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 6617 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4418 . . . 4  |-  ( A 
<Q  B  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
32simpld 110 . . 3  |-  ( A 
<Q  B  ->  A  e. 
Q. )
4 ltexnqi 6661 . . 3  |-  ( A 
<Q  B  ->  E. y  e.  Q.  ( A  +Q  y )  =  B )
5 nsmallnq 6665 . . . . . 6  |-  ( y  e.  Q.  ->  E. z 
z  <Q  y )
61brel 4418 . . . . . . . . . . . . . . 15  |-  ( z 
<Q  y  ->  ( z  e.  Q.  /\  y  e.  Q. ) )
76simpld 110 . . . . . . . . . . . . . 14  |-  ( z 
<Q  y  ->  z  e. 
Q. )
8 ltaddnq 6659 . . . . . . . . . . . . . 14  |-  ( ( A  e.  Q.  /\  z  e.  Q. )  ->  A  <Q  ( A  +Q  z ) )
97, 8sylan2 280 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  z  <Q  y )  ->  A  <Q  ( A  +Q  z ) )
109ancoms 264 . . . . . . . . . . . 12  |-  ( ( z  <Q  y  /\  A  e.  Q. )  ->  A  <Q  ( A  +Q  z ) )
1110adantr 270 . . . . . . . . . . 11  |-  ( ( ( z  <Q  y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  ->  A  <Q  ( A  +Q  z ) )
12 ltanqi 6654 . . . . . . . . . . . . 13  |-  ( ( z  <Q  y  /\  A  e.  Q. )  ->  ( A  +Q  z
)  <Q  ( A  +Q  y ) )
1312adantr 270 . . . . . . . . . . . 12  |-  ( ( ( z  <Q  y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  ->  ( A  +Q  z )  <Q  ( A  +Q  y ) )
14 breq2 3797 . . . . . . . . . . . . 13  |-  ( ( A  +Q  y )  =  B  ->  (
( A  +Q  z
)  <Q  ( A  +Q  y )  <->  ( A  +Q  z )  <Q  B ) )
1514adantl 271 . . . . . . . . . . . 12  |-  ( ( ( z  <Q  y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  ->  ( ( A  +Q  z )  <Q 
( A  +Q  y
)  <->  ( A  +Q  z )  <Q  B ) )
1613, 15mpbid 145 . . . . . . . . . . 11  |-  ( ( ( z  <Q  y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  ->  ( A  +Q  z )  <Q  B )
17 addclnq 6627 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Q.  /\  z  e.  Q. )  ->  ( A  +Q  z
)  e.  Q. )
187, 17sylan2 280 . . . . . . . . . . . . . 14  |-  ( ( A  e.  Q.  /\  z  <Q  y )  -> 
( A  +Q  z
)  e.  Q. )
1918ancoms 264 . . . . . . . . . . . . 13  |-  ( ( z  <Q  y  /\  A  e.  Q. )  ->  ( A  +Q  z
)  e.  Q. )
2019adantr 270 . . . . . . . . . . . 12  |-  ( ( ( z  <Q  y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  ->  ( A  +Q  z )  e.  Q. )
21 breq2 3797 . . . . . . . . . . . . . 14  |-  ( x  =  ( A  +Q  z )  ->  ( A  <Q  x  <->  A  <Q  ( A  +Q  z ) ) )
22 breq1 3796 . . . . . . . . . . . . . 14  |-  ( x  =  ( A  +Q  z )  ->  (
x  <Q  B  <->  ( A  +Q  z )  <Q  B ) )
2321, 22anbi12d 457 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +Q  z )  ->  (
( A  <Q  x  /\  x  <Q  B )  <-> 
( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z
)  <Q  B ) ) )
2423adantl 271 . . . . . . . . . . . 12  |-  ( ( ( ( z  <Q 
y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  /\  x  =  ( A  +Q  z ) )  -> 
( ( A  <Q  x  /\  x  <Q  B )  <-> 
( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z
)  <Q  B ) ) )
2520, 24rspcedv 2706 . . . . . . . . . . 11  |-  ( ( ( z  <Q  y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  ->  ( ( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z )  <Q  B )  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) ) )
2611, 16, 25mp2and 424 . . . . . . . . . 10  |-  ( ( ( z  <Q  y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) )
27263impa 1134 . . . . . . . . 9  |-  ( ( z  <Q  y  /\  A  e.  Q.  /\  ( A  +Q  y )  =  B )  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) )
28273coml 1146 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  ( A  +Q  y
)  =  B  /\  z  <Q  y )  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) )
29283expia 1141 . . . . . . 7  |-  ( ( A  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( z  <Q 
y  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) ) )
3029exlimdv 1741 . . . . . 6  |-  ( ( A  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( E. z 
z  <Q  y  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) ) )
315, 30syl5 32 . . . . 5  |-  ( ( A  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( y  e. 
Q.  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) ) )
3231impancom 256 . . . 4  |-  ( ( A  e.  Q.  /\  y  e.  Q. )  ->  ( ( A  +Q  y )  =  B  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) ) )
3332rexlimdva 2478 . . 3  |-  ( A  e.  Q.  ->  ( E. y  e.  Q.  ( A  +Q  y
)  =  B  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) ) )
343, 4, 33sylc 61 . 2  |-  ( A 
<Q  B  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) )
35 ltsonq 6650 . . . 4  |-  <Q  Or  Q.
3635, 1sotri 4750 . . 3  |-  ( ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
3736rexlimivw 2474 . 2  |-  ( E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
3834, 37impbii 124 1  |-  ( A 
<Q  B  <->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   E.wrex 2350   class class class wbr 3793  (class class class)co 5543   Q.cnq 6532    +Q cplq 6534    <Q cltq 6537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605
This theorem is referenced by:  ltbtwnnq  6668  nqprrnd  6795  appdivnq  6815  ltnqpr  6845  ltnqpri  6846  recexprlemopl  6877  recexprlemopu  6879  cauappcvgprlemopl  6898  cauappcvgprlemopu  6900  cauappcvgprlem2  6912  caucvgprlemopl  6921  caucvgprlemopu  6923  caucvgprlem2  6932
  Copyright terms: Public domain W3C validator