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Theorem cdleme13 30766
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, "<s,t,p> and <f(s),f(t),q> are centrally perspective."  F and  G represent f(s) and f(t) respectively. (Contributed by NM, 7-Oct-2012.)
Hypotheses
Ref Expression
cdleme12.l  |-  .<_  =  ( le `  K )
cdleme12.j  |-  .\/  =  ( join `  K )
cdleme12.m  |-  ./\  =  ( meet `  K )
cdleme12.a  |-  A  =  ( Atoms `  K )
cdleme12.h  |-  H  =  ( LHyp `  K
)
cdleme12.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme12.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme12.g  |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
Assertion
Ref Expression
cdleme13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( S  =/= 
T  /\  -.  U  .<_  ( S  .\/  T
) ) ) )  ->  ( ( S 
.\/  F )  ./\  ( T  .\/  G ) )  .<_  ( P  .\/  Q ) )

Proof of Theorem cdleme13
StepHypRef Expression
1 cdleme12.l . . . 4  |-  .<_  =  ( le `  K )
2 cdleme12.j . . . 4  |-  .\/  =  ( join `  K )
3 cdleme12.m . . . 4  |-  ./\  =  ( meet `  K )
4 cdleme12.a . . . 4  |-  A  =  ( Atoms `  K )
5 cdleme12.h . . . 4  |-  H  =  ( LHyp `  K
)
6 cdleme12.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
7 cdleme12.f . . . 4  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
8 cdleme12.g . . . 4  |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
91, 2, 3, 4, 5, 6, 7, 8cdleme12 30765 . . 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  /\  -.  T  .<_  W )  /\  ( S  =/= 
T  /\  -.  U  .<_  ( S  .\/  T
) ) ) )  ->  ( ( S 
.\/  F )  ./\  ( T  .\/  G ) )  =  U )
109, 6syl6eq 2460 . 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  /\  -.  T  .<_  W )  /\  ( S  =/= 
T  /\  -.  U  .<_  ( S  .\/  T
) ) ) )  ->  ( ( S 
.\/  F )  ./\  ( T  .\/  G ) )  =  ( ( P  .\/  Q ) 
./\  W ) )
11 simp1l 981 . . . 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  /\  -.  T  .<_  W )  /\  ( S  =/= 
T  /\  -.  U  .<_  ( S  .\/  T
) ) ) )  ->  K  e.  HL )
12 hllat 29858 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1311, 12syl 16 . . 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  /\  -.  T  .<_  W )  /\  ( S  =/= 
T  /\  -.  U  .<_  ( S  .\/  T
) ) ) )  ->  K  e.  Lat )
14 simp21l 1074 . . . 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  /\  -.  T  .<_  W )  /\  ( S  =/= 
T  /\  -.  U  .<_  ( S  .\/  T
) ) ) )  ->  P  e.  A
)
15 simp22 991 . . . 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  /\  -.  T  .<_  W )  /\  ( S  =/= 
T  /\  -.  U  .<_  ( S  .\/  T
) ) ) )  ->  Q  e.  A
)
16 eqid 2412 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
1716, 2, 4hlatjcl 29861 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
1811, 14, 15, 17syl3anc 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  /\  -.  T  .<_  W )  /\  ( S  =/= 
T  /\  -.  U  .<_  ( S  .\/  T
) ) ) )  ->  ( P  .\/  Q )  e.  ( Base `  K ) )
19 simp1r 982 . . . 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  /\  -.  T  .<_  W )  /\  ( S  =/= 
T  /\  -.  U  .<_  ( S  .\/  T
) ) ) )  ->  W  e.  H
)
2016, 5lhpbase 30492 . . . 4  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2119, 20syl 16 . . 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  /\  -.  T  .<_  W )  /\  ( S  =/= 
T  /\  -.  U  .<_  ( S  .\/  T
) ) ) )  ->  W  e.  (
Base `  K )
)
2216, 1, 3latmle1 14468 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
2313, 18, 21, 22syl3anc 1184 . 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  /\  -.  T  .<_  W )  /\  ( S  =/= 
T  /\  -.  U  .<_  ( S  .\/  T
) ) ) )  ->  ( ( P 
.\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
2410, 23eqbrtrd 4200 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  /\  -.  T  .<_  W )  /\  ( S  =/= 
T  /\  -.  U  .<_  ( S  .\/  T
) ) ) )  ->  ( ( S 
.\/  F )  ./\  ( T  .\/  G ) )  .<_  ( P  .\/  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   joincjn 14364   meetcmee 14365   Latclat 14437   Atomscatm 29758   HLchlt 29845   LHypclh 30478
This theorem is referenced by:  cdleme14  30767
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482
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