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Theorem cdleme11a 30995
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 31005. (Contributed by NM, 12-Jun-2012.)
Hypotheses
Ref Expression
cdleme11.l  |-  .<_  =  ( le `  K )
cdleme11.j  |-  .\/  =  ( join `  K )
cdleme11.m  |-  ./\  =  ( meet `  K )
cdleme11.a  |-  A  =  ( Atoms `  K )
cdleme11.h  |-  H  =  ( LHyp `  K
)
cdleme11.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme11a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( S  .\/  U
)  =  ( S 
.\/  T ) )

Proof of Theorem cdleme11a
StepHypRef Expression
1 simp3rr 1031 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  U  .<_  ( S  .\/  T ) )
2 simp1l 981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  K  e.  HL )
3 simp1 957 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
4 simp2l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
5 simp2r 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( Q  e.  A  /\  P  =/=  Q
) )
6 cdleme11.l . . . . 5  |-  .<_  =  ( le `  K )
7 cdleme11.j . . . . 5  |-  .\/  =  ( join `  K )
8 cdleme11.m . . . . 5  |-  ./\  =  ( meet `  K )
9 cdleme11.a . . . . 5  |-  A  =  ( Atoms `  K )
10 cdleme11.h . . . . 5  |-  H  =  ( LHyp `  K
)
11 cdleme11.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
126, 7, 8, 9, 10, 11lhpat2 30780 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
133, 4, 5, 12syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  U  e.  A )
14 simp3rl 1030 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  T  e.  A )
15 simp3ll 1028 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  S  e.  A )
16 simp2ll 1024 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  P  e.  A )
17 simp2rl 1026 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  Q  e.  A )
18 simp3l 985 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( S  e.  A  /\  -.  S  .<_  W ) )
196, 7, 8, 9, 10, 11cdleme0c 30948 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  U  =/=  S )
203, 16, 17, 18, 19syl121anc 1189 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  U  =/=  S )
216, 7, 9hlatexchb1 30128 . . 3  |-  ( ( K  e.  HL  /\  ( U  e.  A  /\  T  e.  A  /\  S  e.  A
)  /\  U  =/=  S )  ->  ( U  .<_  ( S  .\/  T
)  <->  ( S  .\/  U )  =  ( S 
.\/  T ) ) )
222, 13, 14, 15, 20, 21syl131anc 1197 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( U  .<_  ( S 
.\/  T )  <->  ( S  .\/  U )  =  ( S  .\/  T ) ) )
231, 22mpbid 202 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( S  .\/  U
)  =  ( S 
.\/  T ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   lecple 13529   joincjn 14394   meetcmee 14395   Atomscatm 29999   HLchlt 30086   LHypclh 30719
This theorem is referenced by:  cdleme11c  30996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-undef 6536  df-riota 6542  df-poset 14396  df-plt 14408  df-lub 14424  df-glb 14425  df-join 14426  df-meet 14427  df-p0 14461  df-p1 14462  df-lat 14468  df-clat 14530  df-oposet 29912  df-ol 29914  df-oml 29915  df-covers 30002  df-ats 30003  df-atl 30034  df-cvlat 30058  df-hlat 30087  df-lhyp 30723
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