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Theorem cdleme35sn3a 30707
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT (Contributed by NM, 19-Mar-2013.)
Hypotheses
Ref Expression
cdleme32s.b  |-  B  =  ( Base `  K
)
cdleme32s.l  |-  .<_  =  ( le `  K )
cdleme32s.j  |-  .\/  =  ( join `  K )
cdleme32s.m  |-  ./\  =  ( meet `  K )
cdleme32s.a  |-  A  =  ( Atoms `  K )
cdleme32s.h  |-  H  =  ( LHyp `  K
)
cdleme32s.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme32s.d  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme32s.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
Assertion
Ref Expression
cdleme35sn3a  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  -.  [_ R  /  s ]_ N  .<_  ( P  .\/  Q ) )
Distinct variable groups:    A, s    B, s    H, s    .\/ , s    K, s    .<_ , s    ./\ , s    P, s    Q, s    R, s    U, s    W, s
Allowed substitution hints:    D( s)    I(
s)    N( s)

Proof of Theorem cdleme35sn3a
StepHypRef Expression
1 cdleme32s.l . . 3  |-  .<_  =  ( le `  K )
2 cdleme32s.j . . 3  |-  .\/  =  ( join `  K )
3 cdleme32s.m . . 3  |-  ./\  =  ( meet `  K )
4 cdleme32s.a . . 3  |-  A  =  ( Atoms `  K )
5 cdleme32s.h . . 3  |-  H  =  ( LHyp `  K
)
6 cdleme32s.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
7 eqid 2366 . . 3  |-  ( ( R  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )  =  ( ( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
81, 2, 3, 4, 5, 6, 7cdleme35fnpq 30697 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  -.  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )  .<_  ( P 
.\/  Q ) )
9 simp2rl 1025 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  R  e.  A )
10 simp3 958 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  -.  R  .<_  ( P  .\/  Q ) )
11 cdleme32s.d . . . . 5  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
12 cdleme32s.n . . . . 5  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
1311, 12, 7cdleme31sn2 30637 . . . 4  |-  ( ( R  e.  A  /\  -.  R  .<_  ( P 
.\/  Q ) )  ->  [_ R  /  s ]_ N  =  (
( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
149, 10, 13syl2anc 642 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  [_ R  /  s ]_ N  =  ( ( R 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) ) )
1514breq1d 4135 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( [_ R  /  s ]_ N  .<_  ( P 
.\/  Q )  <->  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )  .<_  ( P  .\/  Q ) ) )
168, 15mtbird 292 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  -.  [_ R  /  s ]_ N  .<_  ( P  .\/  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   [_csb 3167   ifcif 3654   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   joincjn 14288   meetcmee 14289   Atomscatm 29512   HLchlt 29599   LHypclh 30232
This theorem is referenced by:  cdleme41sn3aw  30722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-lines 29749  df-psubsp 29751  df-pmap 29752  df-padd 30044  df-lhyp 30236
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