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Theorem prcdnq 9367
Description: A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
prcdnq  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  ( C  <Q  B  ->  C  e.  A )
)

Proof of Theorem prcdnq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 9300 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
2 relxp 5108 . . . . . . 7  |-  Rel  ( Q.  X.  Q. )
3 relss 5088 . . . . . . 7  |-  (  <Q  C_  ( Q.  X.  Q. )  ->  ( Rel  ( Q.  X.  Q. )  ->  Rel  <Q  ) )
41, 2, 3mp2 9 . . . . . 6  |-  Rel  <Q
54brrelexi 5039 . . . . 5  |-  ( C 
<Q  B  ->  C  e. 
_V )
6 eleq1 2539 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
76anbi2d 703 . . . . . . . 8  |-  ( x  =  B  ->  (
( A  e.  P.  /\  x  e.  A )  <-> 
( A  e.  P.  /\  B  e.  A ) ) )
8 breq2 4451 . . . . . . . 8  |-  ( x  =  B  ->  (
y  <Q  x  <->  y  <Q  B ) )
97, 8anbi12d 710 . . . . . . 7  |-  ( x  =  B  ->  (
( ( A  e. 
P.  /\  x  e.  A )  /\  y  <Q  x )  <->  ( ( A  e.  P.  /\  B  e.  A )  /\  y  <Q  B ) ) )
109imbi1d 317 . . . . . 6  |-  ( x  =  B  ->  (
( ( ( A  e.  P.  /\  x  e.  A )  /\  y  <Q  x )  ->  y  e.  A )  <->  ( (
( A  e.  P.  /\  B  e.  A )  /\  y  <Q  B )  ->  y  e.  A
) ) )
11 breq1 4450 . . . . . . . 8  |-  ( y  =  C  ->  (
y  <Q  B  <->  C  <Q  B ) )
1211anbi2d 703 . . . . . . 7  |-  ( y  =  C  ->  (
( ( A  e. 
P.  /\  B  e.  A )  /\  y  <Q  B )  <->  ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B ) ) )
13 eleq1 2539 . . . . . . 7  |-  ( y  =  C  ->  (
y  e.  A  <->  C  e.  A ) )
1412, 13imbi12d 320 . . . . . 6  |-  ( y  =  C  ->  (
( ( ( A  e.  P.  /\  B  e.  A )  /\  y  <Q  B )  ->  y  e.  A )  <->  ( (
( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A
) ) )
15 elnpi 9362 . . . . . . . . . . 11  |-  ( A  e.  P.  <->  ( ( A  e.  _V  /\  (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  ( A. y ( y 
<Q  x  ->  y  e.  A )  /\  E. y  e.  A  x  <Q  y ) ) )
1615simprbi 464 . . . . . . . . . 10  |-  ( A  e.  P.  ->  A. x  e.  A  ( A. y ( y  <Q  x  ->  y  e.  A
)  /\  E. y  e.  A  x  <Q  y ) )
1716r19.21bi 2833 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  ( A. y ( y  <Q  x  ->  y  e.  A )  /\  E. y  e.  A  x 
<Q  y ) )
1817simpld 459 . . . . . . . 8  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  A. y ( y 
<Q  x  ->  y  e.  A ) )
191819.21bi 1818 . . . . . . 7  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  ( y  <Q  x  ->  y  e.  A ) )
2019imp 429 . . . . . 6  |-  ( ( ( A  e.  P.  /\  x  e.  A )  /\  y  <Q  x
)  ->  y  e.  A )
2110, 14, 20vtocl2g 3175 . . . . 5  |-  ( ( B  e.  A  /\  C  e.  _V )  ->  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A ) )
225, 21sylan2 474 . . . 4  |-  ( ( B  e.  A  /\  C  <Q  B )  -> 
( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A ) )
2322adantll 713 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A
) )
2423pm2.43i 47 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A
)
2524ex 434 1  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  ( C  <Q  B  ->  C  e.  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476    C. wpss 3477   (/)c0 3785   class class class wbr 4447    X. cxp 4997   Rel wrel 5004   Q.cnq 9226    <Q cltq 9232   P.cnp 9233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-ltnq 9292  df-np 9355
This theorem is referenced by:  prub  9368  addclprlem1  9390  mulclprlem  9393  distrlem4pr  9400  1idpr  9403  psslinpr  9405  prlem934  9407  ltaddpr  9408  ltexprlem2  9411  ltexprlem3  9412  ltexprlem6  9415  prlem936  9421  reclem2pr  9422  suplem1pr  9426
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