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Theorem cdleme43frv1snN 34046
Description: Value of  [_ R  /  s ]_ N when  -.  R  .<_  ( P  .\/  Q ). (Contributed by NM, 30-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemefr27.b  |-  B  =  ( Base `  K
)
cdlemefr27.l  |-  .<_  =  ( le `  K )
cdlemefr27.j  |-  .\/  =  ( join `  K )
cdlemefr27.m  |-  ./\  =  ( meet `  K )
cdlemefr27.a  |-  A  =  ( Atoms `  K )
cdlemefr27.h  |-  H  =  ( LHyp `  K
)
cdlemefr27.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemefr27.c  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdlemefr27.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
cdleme43fr.x  |-  X  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
Assertion
Ref Expression
cdleme43frv1snN  |-  ( ( R  e.  A  /\  -.  R  .<_  ( P 
.\/  Q ) )  ->  [_ R  /  s ]_ N  =  X
)
Distinct variable groups:    A, s    .\/ , s    .<_ , s    ./\ , s    P, s    Q, s    R, s    U, s    W, s    H, s    K, s    B, s
Allowed substitution hints:    C( s)    I(
s)    N( s)    X( s)

Proof of Theorem cdleme43frv1snN
StepHypRef Expression
1 cdlemefr27.c . 2  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
2 cdlemefr27.n . 2  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
3 cdleme43fr.x . 2  |-  X  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
41, 2, 3cdleme31sn2 34027 1  |-  ( ( R  e.  A  /\  -.  R  .<_  ( P 
.\/  Q ) )  ->  [_ R  /  s ]_ N  =  X
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   [_csb 3349   ifcif 3872   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   meetcmee 16268   Atomscatm 32900   LHypclh 33620
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-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  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-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-ov 6311
This theorem is referenced by: (None)
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