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Theorem ltbtwnnqq 6513
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 6463 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4392 . . . 4  |-  ( A 
<Q  B  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
32simpld 105 . . 3  |-  ( A 
<Q  B  ->  A  e. 
Q. )
4 ltexnqi 6507 . . 3  |-  ( A 
<Q  B  ->  E. y  e.  Q.  ( A  +Q  y )  =  B )
5 nsmallnq 6511 . . . . . 6  |-  ( y  e.  Q.  ->  E. z 
z  <Q  y )
61brel 4392 . . . . . . . . . . . . . . 15  |-  ( z 
<Q  y  ->  ( z  e.  Q.  /\  y  e.  Q. ) )
76simpld 105 . . . . . . . . . . . . . 14  |-  ( z 
<Q  y  ->  z  e. 
Q. )
8 ltaddnq 6505 . . . . . . . . . . . . . 14  |-  ( ( A  e.  Q.  /\  z  e.  Q. )  ->  A  <Q  ( A  +Q  z ) )
97, 8sylan2 270 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  z  <Q  y )  ->  A  <Q  ( A  +Q  z ) )
109ancoms 255 . . . . . . . . . . . 12  |-  ( ( z  <Q  y  /\  A  e.  Q. )  ->  A  <Q  ( A  +Q  z ) )
1110adantr 261 . . . . . . . . . . 11  |-  ( ( ( z  <Q  y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  ->  A  <Q  ( A  +Q  z ) )
12 ltanqi 6500 . . . . . . . . . . . . 13  |-  ( ( z  <Q  y  /\  A  e.  Q. )  ->  ( A  +Q  z
)  <Q  ( A  +Q  y ) )
1312adantr 261 . . . . . . . . . . . 12  |-  ( ( ( z  <Q  y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  ->  ( A  +Q  z )  <Q  ( A  +Q  y ) )
14 breq2 3768 . . . . . . . . . . . . 13  |-  ( ( A  +Q  y )  =  B  ->  (
( A  +Q  z
)  <Q  ( A  +Q  y )  <->  ( A  +Q  z )  <Q  B ) )
1514adantl 262 . . . . . . . . . . . 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 135 . . . . . . . . . . 11  |-  ( ( ( z  <Q  y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  ->  ( A  +Q  z )  <Q  B )
17 addclnq 6473 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  Q.  /\  z  e.  Q. )  ->  ( A  +Q  z
)  e.  Q. )
187, 17sylan2 270 . . . . . . . . . . . . . 14  |-  ( ( A  e.  Q.  /\  z  <Q  y )  -> 
( A  +Q  z
)  e.  Q. )
1918ancoms 255 . . . . . . . . . . . . 13  |-  ( ( z  <Q  y  /\  A  e.  Q. )  ->  ( A  +Q  z
)  e.  Q. )
2019adantr 261 . . . . . . . . . . . 12  |-  ( ( ( z  <Q  y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  ->  ( A  +Q  z )  e.  Q. )
21 breq2 3768 . . . . . . . . . . . . . 14  |-  ( x  =  ( A  +Q  z )  ->  ( A  <Q  x  <->  A  <Q  ( A  +Q  z ) ) )
22 breq1 3767 . . . . . . . . . . . . . 14  |-  ( x  =  ( A  +Q  z )  ->  (
x  <Q  B  <->  ( A  +Q  z )  <Q  B ) )
2321, 22anbi12d 442 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +Q  z )  ->  (
( A  <Q  x  /\  x  <Q  B )  <-> 
( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z
)  <Q  B ) ) )
2423adantl 262 . . . . . . . . . . . 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 2660 . . . . . . . . . . 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 409 . . . . . . . . . 10  |-  ( ( ( z  <Q  y  /\  A  e.  Q. )  /\  ( A  +Q  y )  =  B )  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) )
27263impa 1099 . . . . . . . . 9  |-  ( ( z  <Q  y  /\  A  e.  Q.  /\  ( A  +Q  y )  =  B )  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) )
28273coml 1111 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  ( A  +Q  y
)  =  B  /\  z  <Q  y )  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) )
29283expia 1106 . . . . . . 7  |-  ( ( A  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( z  <Q 
y  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) ) )
3029exlimdv 1700 . . . . . 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 28 . . . . 5  |-  ( ( A  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( y  e. 
Q.  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) ) )
3231impancom 247 . . . 4  |-  ( ( A  e.  Q.  /\  y  e.  Q. )  ->  ( ( A  +Q  y )  =  B  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) ) )
3332rexlimdva 2433 . . 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 56 . 2  |-  ( A 
<Q  B  ->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) )
35 ltsonq 6496 . . . 4  |-  <Q  Or  Q.
3635, 1sotri 4720 . . 3  |-  ( ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
3736rexlimivw 2429 . 2  |-  ( E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
3834, 37impbii 117 1  |-  ( A 
<Q  B  <->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393   E.wrex 2307   class class class wbr 3764  (class class class)co 5512   Q.cnq 6378    +Q cplq 6380    <Q cltq 6383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451
This theorem is referenced by:  ltbtwnnq  6514  nqprrnd  6641  appdivnq  6661  ltnqpr  6691  ltnqpri  6692  recexprlemopl  6723  recexprlemopu  6725  cauappcvgprlemopl  6744  cauappcvgprlemopu  6746  cauappcvgprlem2  6758  caucvgprlemopl  6767  caucvgprlemopu  6769  caucvgprlem2  6778
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