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Theorem cdleme21a 30807
Description: Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
Hypotheses
Ref Expression
cdleme21a.l  |-  .<_  =  ( le `  K )
cdleme21a.j  |-  .\/  =  ( join `  K )
cdleme21a.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cdleme21a  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) )  /\  ( z  e.  A  /\  ( P 
.\/  z )  =  ( S  .\/  z
) ) )  ->  S  =/=  z )

Proof of Theorem cdleme21a
StepHypRef Expression
1 simp11 987 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) )  /\  ( z  e.  A  /\  ( P 
.\/  z )  =  ( S  .\/  z
) ) )  ->  K  e.  HL )
2 hlcvl 29842 . . 3  |-  ( K  e.  HL  ->  K  e.  CvLat )
31, 2syl 16 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) )  /\  ( z  e.  A  /\  ( P 
.\/  z )  =  ( S  .\/  z
) ) )  ->  K  e.  CvLat )
4 simp12 988 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) )  /\  ( z  e.  A  /\  ( P 
.\/  z )  =  ( S  .\/  z
) ) )  ->  P  e.  A )
5 simp2l 983 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) )  /\  ( z  e.  A  /\  ( P 
.\/  z )  =  ( S  .\/  z
) ) )  ->  S  e.  A )
6 simp3l 985 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) )  /\  ( z  e.  A  /\  ( P 
.\/  z )  =  ( S  .\/  z
) ) )  -> 
z  e.  A )
7 simp13 989 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) )  /\  ( z  e.  A  /\  ( P 
.\/  z )  =  ( S  .\/  z
) ) )  ->  Q  e.  A )
8 simp2r 984 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) )  /\  ( z  e.  A  /\  ( P 
.\/  z )  =  ( S  .\/  z
) ) )  ->  -.  S  .<_  ( P 
.\/  Q ) )
9 cdleme21a.l . . . . 5  |-  .<_  =  ( le `  K )
10 cdleme21a.j . . . . 5  |-  .\/  =  ( join `  K )
11 cdleme21a.a . . . . 5  |-  A  =  ( Atoms `  K )
129, 10, 11atnlej1 29861 . . . 4  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  S  =/=  P )
1312necomd 2650 . . 3  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  =/=  S )
141, 5, 4, 7, 8, 13syl131anc 1197 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) )  /\  ( z  e.  A  /\  ( P 
.\/  z )  =  ( S  .\/  z
) ) )  ->  P  =/=  S )
15 simp3r 986 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) )  /\  ( z  e.  A  /\  ( P 
.\/  z )  =  ( S  .\/  z
) ) )  -> 
( P  .\/  z
)  =  ( S 
.\/  z ) )
1611, 10cvlsupr6 29830 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  S  e.  A  /\  z  e.  A )  /\  ( P  =/=  S  /\  ( P  .\/  z
)  =  ( S 
.\/  z ) ) )  ->  z  =/=  S )
1716necomd 2650 . 2  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  S  e.  A  /\  z  e.  A )  /\  ( P  =/=  S  /\  ( P  .\/  z
)  =  ( S 
.\/  z ) ) )  ->  S  =/=  z )
183, 4, 5, 6, 14, 15, 17syl132anc 1202 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) )  /\  ( z  e.  A  /\  ( P 
.\/  z )  =  ( S  .\/  z
) ) )  ->  S  =/=  z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   Atomscatm 29746   CvLatclc 29748   HLchlt 29833
This theorem is referenced by:  cdleme21ct  30811  cdleme21d  30812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-join 14388  df-lat 14430  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834
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