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Theorem cdleme0c 29652
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)
Hypotheses
Ref Expression
cdleme0.l  |-  .<_  =  ( le `  K )
cdleme0.j  |-  .\/  =  ( join `  K )
cdleme0.m  |-  ./\  =  ( meet `  K )
cdleme0.a  |-  A  =  ( Atoms `  K )
cdleme0.h  |-  H  =  ( LHyp `  K
)
cdleme0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme0c  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  U  =/=  R )

Proof of Theorem cdleme0c
StepHypRef Expression
1 cdleme0.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2 simp1l 984 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  K  e.  HL )
3 hllat 28803 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  K  e.  Lat )
5 simp2l 986 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  P  e.  A )
6 eqid 2258 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
86, 7atbase 28729 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
95, 8syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  P  e.  ( Base `  K )
)
10 simp2r 987 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  Q  e.  A )
116, 7atbase 28729 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1210, 11syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  Q  e.  ( Base `  K )
)
13 cdleme0.j . . . . . 6  |-  .\/  =  ( join `  K )
146, 13latjcl 14119 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
154, 9, 12, 14syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
16 simp1r 985 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  W  e.  H )
17 cdleme0.h . . . . . 6  |-  H  =  ( LHyp `  K
)
186, 17lhpbase 29437 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1916, 18syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  W  e.  ( Base `  K )
)
20 cdleme0.l . . . . 5  |-  .<_  =  ( le `  K )
21 cdleme0.m . . . . 5  |-  ./\  =  ( meet `  K )
226, 20, 21latmle2 14146 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
234, 15, 19, 22syl3anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
241, 23syl5eqbr 4030 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  U  .<_  W )
25 simp3r 989 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  -.  R  .<_  W )
26 nbrne2 4015 . 2  |-  ( ( U  .<_  W  /\  -.  R  .<_  W )  ->  U  =/=  R
)
2724, 25, 26syl2anc 645 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  U  =/=  R )
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   Basecbs 13111   lecple 13178   joincjn 14041   meetcmee 14042   Latclat 14114   Atomscatm 28703   HLchlt 28790   LHypclh 29423
This theorem is referenced by:  cdleme0gN  29658  cdleme11a  29699  cdleme11h  29705  cdleme36a  29899
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-glb 14072  df-meet 14074  df-lat 14115  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-lhyp 29427
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