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Theorem prcdnq 8613
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 8546 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
2 relxp 4793 . . . . . . 7  |-  Rel  ( Q.  X.  Q. )
3 relss 4774 . . . . . . 7  |-  (  <Q  C_  ( Q.  X.  Q. )  ->  ( Rel  ( Q.  X.  Q. )  ->  Rel  <Q  ) )
41, 2, 3mp2 17 . . . . . 6  |-  Rel  <Q
54brrelexi 4728 . . . . 5  |-  ( C 
<Q  B  ->  C  e. 
_V )
6 eleq1 2344 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
76anbi2d 684 . . . . . . . 8  |-  ( x  =  B  ->  (
( A  e.  P.  /\  x  e.  A )  <-> 
( A  e.  P.  /\  B  e.  A ) ) )
8 breq2 4028 . . . . . . . 8  |-  ( x  =  B  ->  (
y  <Q  x  <->  y  <Q  B ) )
97, 8anbi12d 691 . . . . . . 7  |-  ( x  =  B  ->  (
( ( A  e. 
P.  /\  x  e.  A )  /\  y  <Q  x )  <->  ( ( A  e.  P.  /\  B  e.  A )  /\  y  <Q  B ) ) )
109imbi1d 308 . . . . . 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 4027 . . . . . . . 8  |-  ( y  =  C  ->  (
y  <Q  B  <->  C  <Q  B ) )
1211anbi2d 684 . . . . . . 7  |-  ( y  =  C  ->  (
( ( A  e. 
P.  /\  B  e.  A )  /\  y  <Q  B )  <->  ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B ) ) )
13 eleq1 2344 . . . . . . 7  |-  ( y  =  C  ->  (
y  e.  A  <->  C  e.  A ) )
1412, 13imbi12d 311 . . . . . 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 8608 . . . . . . . . . . 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 450 . . . . . . . . . 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 2642 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  ( A. y ( y  <Q  x  ->  y  e.  A )  /\  E. y  e.  A  x 
<Q  y ) )
1817simpld 445 . . . . . . . 8  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  A. y ( y 
<Q  x  ->  y  e.  A ) )
191819.21bi 1796 . . . . . . 7  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  ( y  <Q  x  ->  y  e.  A ) )
2019imp 418 . . . . . 6  |-  ( ( ( A  e.  P.  /\  x  e.  A )  /\  y  <Q  x
)  ->  y  e.  A )
2110, 14, 20vtocl2g 2848 . . . . 5  |-  ( ( B  e.  A  /\  C  e.  _V )  ->  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A ) )
225, 21sylan2 460 . . . 4  |-  ( ( B  e.  A  /\  C  <Q  B )  -> 
( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A ) )
2322adantll 694 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A
) )
2423pm2.43i 43 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A
)
2524ex 423 1  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  ( C  <Q  B  ->  C  e.  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1685   A.wral 2544   E.wrex 2545   _Vcvv 2789    C_ wss 3153    C. wpss 3154   (/)c0 3456   class class class wbr 4024    X. cxp 4686   Rel wrel 4693   Q.cnq 8470    <Q cltq 8476   P.cnp 8477
This theorem is referenced by:  prub  8614  addclprlem1  8636  mulclprlem  8639  distrlem4pr  8646  1idpr  8649  psslinpr  8651  prlem934  8653  ltaddpr  8654  ltexprlem2  8657  ltexprlem3  8658  ltexprlem6  8661  prlem936  8667  reclem2pr  8668  suplem1pr  8672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-ltnq 8538  df-np 8601
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