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Theorem ltbtwnnq 9234
Description: There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltbtwnnq  |-  ( A 
<Q  B  <->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem ltbtwnnq
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 9182 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4971 . . . 4  |-  ( A 
<Q  B  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
32simprd 463 . . 3  |-  ( A 
<Q  B  ->  B  e. 
Q. )
4 ltexnq 9231 . . . 4  |-  ( B  e.  Q.  ->  ( A  <Q  B  <->  E. y
( A  +Q  y
)  =  B ) )
5 eleq1 2520 . . . . . . . . . 10  |-  ( ( A  +Q  y )  =  B  ->  (
( A  +Q  y
)  e.  Q.  <->  B  e.  Q. ) )
65biimparc 487 . . . . . . . . 9  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( A  +Q  y )  e.  Q. )
7 addnqf 9204 . . . . . . . . . . 11  |-  +Q  :
( Q.  X.  Q. )
--> Q.
87fdmi 5648 . . . . . . . . . 10  |-  dom  +Q  =  ( Q.  X.  Q. )
9 0nnq 9180 . . . . . . . . . 10  |-  -.  (/)  e.  Q.
108, 9ndmovrcl 6335 . . . . . . . . 9  |-  ( ( A  +Q  y )  e.  Q.  ->  ( A  e.  Q.  /\  y  e.  Q. ) )
116, 10syl 16 . . . . . . . 8  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( A  e. 
Q.  /\  y  e.  Q. ) )
1211simprd 463 . . . . . . 7  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  y  e.  Q. )
13 nsmallnq 9233 . . . . . . . 8  |-  ( y  e.  Q.  ->  E. z 
z  <Q  y )
1411simpld 459 . . . . . . . . . . . 12  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  A  e.  Q. )
151brel 4971 . . . . . . . . . . . . 13  |-  ( z 
<Q  y  ->  ( z  e.  Q.  /\  y  e.  Q. ) )
1615simpld 459 . . . . . . . . . . . 12  |-  ( z 
<Q  y  ->  z  e. 
Q. )
17 ltaddnq 9230 . . . . . . . . . . . 12  |-  ( ( A  e.  Q.  /\  z  e.  Q. )  ->  A  <Q  ( A  +Q  z ) )
1814, 16, 17syl2an 477 . . . . . . . . . . 11  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  A  <Q  ( A  +Q  z ) )
19 ltanq 9227 . . . . . . . . . . . . . 14  |-  ( A  e.  Q.  ->  (
z  <Q  y  <->  ( A  +Q  z )  <Q  ( A  +Q  y ) ) )
2019biimpa 484 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  z  <Q  y )  -> 
( A  +Q  z
)  <Q  ( A  +Q  y ) )
2114, 20sylan 471 . . . . . . . . . . . 12  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  z )  <Q  ( A  +Q  y ) )
22 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  y )  =  B )
2321, 22breqtrd 4400 . . . . . . . . . . 11  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  z )  <Q  B )
24 ovex 6201 . . . . . . . . . . . 12  |-  ( A  +Q  z )  e. 
_V
25 breq2 4380 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +Q  z )  ->  ( A  <Q  x  <->  A  <Q  ( A  +Q  z ) ) )
26 breq1 4379 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +Q  z )  ->  (
x  <Q  B  <->  ( A  +Q  z )  <Q  B ) )
2725, 26anbi12d 710 . . . . . . . . . . . 12  |-  ( x  =  ( A  +Q  z )  ->  (
( A  <Q  x  /\  x  <Q  B )  <-> 
( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z
)  <Q  B ) ) )
2824, 27spcev 3146 . . . . . . . . . . 11  |-  ( ( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z )  <Q  B )  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
2918, 23, 28syl2anc 661 . . . . . . . . . 10  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
3029ex 434 . . . . . . . . 9  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( z  <Q 
y  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
3130exlimdv 1691 . . . . . . . 8  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( E. z 
z  <Q  y  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
3213, 31syl5 32 . . . . . . 7  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( y  e. 
Q.  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) ) )
3312, 32mpd 15 . . . . . 6  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
3433ex 434 . . . . 5  |-  ( B  e.  Q.  ->  (
( A  +Q  y
)  =  B  ->  E. x ( A  <Q  x  /\  x  <Q  B ) ) )
3534exlimdv 1691 . . . 4  |-  ( B  e.  Q.  ->  ( E. y ( A  +Q  y )  =  B  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) ) )
364, 35sylbid 215 . . 3  |-  ( B  e.  Q.  ->  ( A  <Q  B  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
373, 36mpcom 36 . 2  |-  ( A 
<Q  B  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
38 ltsonq 9225 . . . 4  |-  <Q  Or  Q.
3938, 1sotri 5309 . . 3  |-  ( ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
4039exlimiv 1689 . 2  |-  ( E. x ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
4137, 40impbii 188 1  |-  ( A 
<Q  B  <->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1757   class class class wbr 4376    X. cxp 4922  (class class class)co 6176   Q.cnq 9106    +Q cplq 9109    <Q cltq 9112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-omul 7011  df-er 7187  df-ni 9128  df-pli 9129  df-mi 9130  df-lti 9131  df-plpq 9164  df-mpq 9165  df-ltpq 9166  df-enq 9167  df-nq 9168  df-erq 9169  df-plq 9170  df-mq 9171  df-1nq 9172  df-rq 9173  df-ltnq 9174
This theorem is referenced by:  nqpr  9270  reclem2pr  9304
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