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Theorem cdleme11g 33843
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 33848. (Contributed by NM, 14-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 )
cdleme11.c  |-  C  =  ( ( P  .\/  S )  ./\  W )
cdleme11.d  |-  D  =  ( ( P  .\/  T )  ./\  W )
cdleme11.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme11g  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  F )  =  ( Q  .\/  C ) )

Proof of Theorem cdleme11g
StepHypRef Expression
1 cdleme11.f . . . 4  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
21oveq2i 6306 . . 3  |-  ( Q 
.\/  F )  =  ( Q  .\/  (
( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
3 simp1l 1033 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
4 simp22l 1128 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
5 hllat 32941 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
63, 5syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  K  e.  Lat )
7 simp23 1044 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  S  e.  A )
8 eqid 2453 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
9 cdleme11.a . . . . . . 7  |-  A  =  ( Atoms `  K )
108, 9atbase 32867 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
117, 10syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  S  e.  ( Base `  K )
)
12 simp1 1009 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( K  e.  HL  /\  W  e.  H ) )
13 simp21 1042 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
14 cdleme11.l . . . . . . 7  |-  .<_  =  ( le `  K )
15 cdleme11.j . . . . . . 7  |-  .\/  =  ( join `  K )
16 cdleme11.m . . . . . . 7  |-  ./\  =  ( meet `  K )
17 cdleme11.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
18 cdleme11.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
1914, 15, 16, 9, 17, 18, 8cdleme0aa 33788 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  U  e.  ( Base `  K )
)
2012, 13, 4, 19syl3anc 1269 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  U  e.  ( Base `  K )
)
218, 15latjcl 16309 . . . . 5  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  ( S  .\/  U )  e.  ( Base `  K
) )
226, 11, 20, 21syl3anc 1269 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( S  .\/  U )  e.  (
Base `  K )
)
238, 9atbase 32867 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
244, 23syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  ( Base `  K )
)
258, 9atbase 32867 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2613, 25syl 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  P  e.  ( Base `  K )
)
278, 15latjcl 16309 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  ( P  .\/  S )  e.  ( Base `  K
) )
286, 26, 11, 27syl3anc 1269 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  S )  e.  (
Base `  K )
)
29 simp1r 1034 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  W  e.  H )
308, 17lhpbase 33575 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3129, 30syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  W  e.  ( Base `  K )
)
328, 16latmcl 16310 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  e.  ( Base `  K ) )
336, 28, 31, 32syl3anc 1269 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  S )  ./\  W )  e.  ( Base `  K ) )
348, 15latjcl 16309 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( P  .\/  S
)  ./\  W )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)
356, 24, 33, 34syl3anc 1269 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
)  e.  ( Base `  K ) )
368, 14, 15latlej1 16318 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( P  .\/  S
)  ./\  W )  e.  ( Base `  K
) )  ->  Q  .<_  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) )
376, 24, 33, 36syl3anc 1269 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  Q  .<_  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) ) )
388, 14, 15, 16, 9atmod1i1 33434 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  ( S  .\/  U
)  e.  ( Base `  K )  /\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)  /\  Q  .<_  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) ) )  ->  ( Q  .\/  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )  =  ( ( Q  .\/  ( S  .\/  U ) )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
393, 4, 22, 35, 37, 38syl131anc 1282 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )  =  ( ( Q  .\/  ( S  .\/  U ) )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
402, 39syl5eq 2499 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  F )  =  ( ( Q  .\/  ( S  .\/  U ) ) 
./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
41 simp22 1043 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
4214, 15, 16, 9, 17, 18cdleme0cq 33793 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( Q  .\/  U )  =  ( P  .\/  Q ) )
4312, 13, 41, 42syl12anc 1267 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  U )  =  ( P  .\/  Q ) )
4443oveq2d 6311 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( S  .\/  ( Q  .\/  U
) )  =  ( S  .\/  ( P 
.\/  Q ) ) )
458, 15latj12 16354 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  S  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) ) )  -> 
( Q  .\/  ( S  .\/  U ) )  =  ( S  .\/  ( Q  .\/  U ) ) )
466, 24, 11, 20, 45syl13anc 1271 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( S  .\/  U
) )  =  ( S  .\/  ( Q 
.\/  U ) ) )
478, 15latj13 16356 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) ) )  -> 
( Q  .\/  ( P  .\/  S ) )  =  ( S  .\/  ( P  .\/  Q ) ) )
486, 24, 26, 11, 47syl13anc 1271 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( P  .\/  S
) )  =  ( S  .\/  ( P 
.\/  Q ) ) )
4944, 46, 483eqtr4d 2497 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( S  .\/  U
) )  =  ( Q  .\/  ( P 
.\/  S ) ) )
5049oveq1d 6310 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( Q  .\/  ( S  .\/  U ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
518, 14, 16latmle1 16334 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
526, 28, 31, 51syl3anc 1269 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
538, 14, 15latjlej2 16324 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( ( P 
.\/  S )  ./\  W )  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S
)  ->  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
)  .<_  ( Q  .\/  ( P  .\/  S ) ) ) )
546, 33, 28, 24, 53syl13anc 1271 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( (
( P  .\/  S
)  ./\  W )  .<_  ( P  .\/  S
)  ->  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
)  .<_  ( Q  .\/  ( P  .\/  S ) ) ) )
5552, 54mpd 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
)  .<_  ( Q  .\/  ( P  .\/  S ) ) )
568, 15latjcl 16309 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( P  .\/  S ) )  e.  (
Base `  K )
)
576, 24, 28, 56syl3anc 1269 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( P  .\/  S
) )  e.  (
Base `  K )
)
588, 14, 16latleeqm2 16338 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) )  e.  ( Base `  K
)  /\  ( Q  .\/  ( P  .\/  S
) )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) )  .<_  ( Q 
.\/  ( P  .\/  S ) )  <->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) ) )
596, 35, 57, 58syl3anc 1269 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) )  .<_  ( Q 
.\/  ( P  .\/  S ) )  <->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) ) )
6055, 59mpbid 214 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) )
61 cdleme11.c . . . 4  |-  C  =  ( ( P  .\/  S )  ./\  W )
6261oveq2i 6306 . . 3  |-  ( Q 
.\/  C )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
)
6360, 62syl6eqr 2505 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( Q  .\/  C
) )
6440, 50, 633eqtrd 2491 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  F )  =  ( Q  .\/  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   Basecbs 15133   lecple 15209   joincjn 16201   meetcmee 16202   Latclat 16303   Atomscatm 32841   HLchlt 32928   LHypclh 33561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  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 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-preset 16185  df-poset 16203  df-plt 16216  df-lub 16232  df-glb 16233  df-join 16234  df-meet 16235  df-p0 16297  df-p1 16298  df-lat 16304  df-clat 16366  df-oposet 32754  df-ol 32756  df-oml 32757  df-covers 32844  df-ats 32845  df-atl 32876  df-cvlat 32900  df-hlat 32929  df-psubsp 33080  df-pmap 33081  df-padd 33373  df-lhyp 33565
This theorem is referenced by:  cdleme11h  33844  cdleme11j  33845  cdleme15a  33852
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