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Theorem ltbtwnnq 9421
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 9369 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4888 . . . 4  |-  ( A 
<Q  B  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
32simprd 470 . . 3  |-  ( A 
<Q  B  ->  B  e. 
Q. )
4 ltexnq 9418 . . . 4  |-  ( B  e.  Q.  ->  ( A  <Q  B  <->  E. y
( A  +Q  y
)  =  B ) )
5 eleq1 2537 . . . . . . . . . 10  |-  ( ( A  +Q  y )  =  B  ->  (
( A  +Q  y
)  e.  Q.  <->  B  e.  Q. ) )
65biimparc 495 . . . . . . . . 9  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( A  +Q  y )  e.  Q. )
7 addnqf 9391 . . . . . . . . . . 11  |-  +Q  :
( Q.  X.  Q. )
--> Q.
87fdmi 5746 . . . . . . . . . 10  |-  dom  +Q  =  ( Q.  X.  Q. )
9 0nnq 9367 . . . . . . . . . 10  |-  -.  (/)  e.  Q.
108, 9ndmovrcl 6474 . . . . . . . . 9  |-  ( ( A  +Q  y )  e.  Q.  ->  ( A  e.  Q.  /\  y  e.  Q. ) )
116, 10syl 17 . . . . . . . 8  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( A  e. 
Q.  /\  y  e.  Q. ) )
1211simprd 470 . . . . . . 7  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  y  e.  Q. )
13 nsmallnq 9420 . . . . . . . 8  |-  ( y  e.  Q.  ->  E. z 
z  <Q  y )
1411simpld 466 . . . . . . . . . . . 12  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  A  e.  Q. )
151brel 4888 . . . . . . . . . . . . 13  |-  ( z 
<Q  y  ->  ( z  e.  Q.  /\  y  e.  Q. ) )
1615simpld 466 . . . . . . . . . . . 12  |-  ( z 
<Q  y  ->  z  e. 
Q. )
17 ltaddnq 9417 . . . . . . . . . . . 12  |-  ( ( A  e.  Q.  /\  z  e.  Q. )  ->  A  <Q  ( A  +Q  z ) )
1814, 16, 17syl2an 485 . . . . . . . . . . 11  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  A  <Q  ( A  +Q  z ) )
19 ltanq 9414 . . . . . . . . . . . . . 14  |-  ( A  e.  Q.  ->  (
z  <Q  y  <->  ( A  +Q  z )  <Q  ( A  +Q  y ) ) )
2019biimpa 492 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  z  <Q  y )  -> 
( A  +Q  z
)  <Q  ( A  +Q  y ) )
2114, 20sylan 479 . . . . . . . . . . . 12  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  z )  <Q  ( A  +Q  y ) )
22 simplr 770 . . . . . . . . . . . 12  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  y )  =  B )
2321, 22breqtrd 4420 . . . . . . . . . . 11  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  z )  <Q  B )
24 ovex 6336 . . . . . . . . . . . 12  |-  ( A  +Q  z )  e. 
_V
25 breq2 4399 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +Q  z )  ->  ( A  <Q  x  <->  A  <Q  ( A  +Q  z ) ) )
26 breq1 4398 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +Q  z )  ->  (
x  <Q  B  <->  ( A  +Q  z )  <Q  B ) )
2725, 26anbi12d 725 . . . . . . . . . . . 12  |-  ( x  =  ( A  +Q  z )  ->  (
( A  <Q  x  /\  x  <Q  B )  <-> 
( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z
)  <Q  B ) ) )
2824, 27spcev 3127 . . . . . . . . . . 11  |-  ( ( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z )  <Q  B )  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
2918, 23, 28syl2anc 673 . . . . . . . . . 10  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
3029ex 441 . . . . . . . . 9  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( z  <Q 
y  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
3130exlimdv 1787 . . . . . . . 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 441 . . . . 5  |-  ( B  e.  Q.  ->  (
( A  +Q  y
)  =  B  ->  E. x ( A  <Q  x  /\  x  <Q  B ) ) )
3534exlimdv 1787 . . . 4  |-  ( B  e.  Q.  ->  ( E. y ( A  +Q  y )  =  B  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) ) )
364, 35sylbid 223 . . 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 9412 . . . 4  |-  <Q  Or  Q.
3938, 1sotri 5233 . . 3  |-  ( ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
4039exlimiv 1784 . 2  |-  ( E. x ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
4137, 40impbii 192 1  |-  ( A 
<Q  B  <->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   class class class wbr 4395    X. cxp 4837  (class class class)co 6308   Q.cnq 9295    +Q cplq 9298    <Q cltq 9301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-omul 7205  df-er 7381  df-ni 9315  df-pli 9316  df-mi 9317  df-lti 9318  df-plpq 9351  df-mpq 9352  df-ltpq 9353  df-enq 9354  df-nq 9355  df-erq 9356  df-plq 9357  df-mq 9358  df-1nq 9359  df-rq 9360  df-ltnq 9361
This theorem is referenced by:  nqpr  9457  reclem2pr  9491
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