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Theorem cdleme21a 29681
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 990 . . 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 28716 . . 3  |-  ( K  e.  HL  ->  K  e.  CvLat )
31, 2syl 17 . 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 991 . 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 986 . 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 988 . 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 992 . . 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 987 . . 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 28735 . . . 4  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  S  =/=  P )
1312necomd 2504 . . 3  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  =/=  S )
141, 5, 4, 7, 8, 13syl131anc 1200 . 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 989 . 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 28704 . . 3  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  S  e.  A  /\  z  e.  A )  /\  ( P  =/=  S  /\  ( P  .\/  z
)  =  ( S 
.\/  z ) ) )  ->  z  =/=  S )
1716necomd 2504 . 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 1205 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   lecple 13177   joincjn 14040   Atomscatm 28620   CvLatclc 28622   HLchlt 28707
This theorem is referenced by:  cdleme21ct  29685  cdleme21d  29686
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-plt 14054  df-lub 14070  df-join 14072  df-lat 14114  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708
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