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Theorem cdleme11e 30428
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30435. (Contributed by NM, 13-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 )
Assertion
Ref Expression
cdleme11e  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  C  =/=  D )

Proof of Theorem cdleme11e
StepHypRef Expression
1 simp11 987 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 988 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp22 991 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  T  e.  A )
4 simp21 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
5 simp11l 1068 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  K  e.  HL )
6 hllat 29529 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
75, 6syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  K  e.  Lat )
8 simp12l 1070 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  P  e.  A )
9 eqid 2380 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
10 cdleme11.a . . . . . 6  |-  A  =  ( Atoms `  K )
119, 10atbase 29455 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
128, 11syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  P  e.  ( Base `  K
) )
13 simp21l 1074 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  S  e.  A )
149, 10atbase 29455 . . . . 5  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1513, 14syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  S  e.  ( Base `  K
) )
169, 10atbase 29455 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
173, 16syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  T  e.  ( Base `  K
) )
18 simp1 957 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A ) )
19 simp2 958 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  (
( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
) )
20 simp32 994 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
21 simp33 995 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  U  .<_  ( S  .\/  T
) )
22 cdleme11.l . . . . . 6  |-  .<_  =  ( le `  K )
23 cdleme11.j . . . . . 6  |-  .\/  =  ( join `  K )
24 cdleme11.m . . . . . 6  |-  ./\  =  ( meet `  K )
25 cdleme11.h . . . . . 6  |-  H  =  ( LHyp `  K
)
26 cdleme11.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2722, 23, 24, 10, 25, 26cdleme11c 30426 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  U  .<_  ( S  .\/  T ) ) )  ->  -.  P  .<_  ( S  .\/  T ) )
2818, 19, 20, 21, 27syl112anc 1188 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  -.  P  .<_  ( S  .\/  T ) )
299, 22, 23latnlej1r 14419 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  S  e.  ( Base `  K )  /\  T  e.  ( Base `  K
) )  /\  -.  P  .<_  ( S  .\/  T ) )  ->  P  =/=  T )
307, 12, 15, 17, 28, 29syl131anc 1197 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  P  =/=  T )
31 simp31 993 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  S  =/=  T )
3222, 23, 10hlatcon2 29617 . . . 4  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  P  e.  A
)  /\  ( S  =/=  T  /\  -.  P  .<_  ( S  .\/  T
) ) )  ->  -.  S  .<_  ( P 
.\/  T ) )
335, 13, 3, 8, 31, 28, 32syl132anc 1202 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  -.  S  .<_  ( P  .\/  T ) )
34 cdleme11.d . . . 4  |-  D  =  ( ( P  .\/  T )  ./\  W )
35 cdleme11.c . . . 4  |-  C  =  ( ( P  .\/  S )  ./\  W )
3622, 23, 24, 10, 25, 34, 35cdleme0e 30382 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  T  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  T  /\  -.  S  .<_  ( P  .\/  T ) ) )  ->  D  =/=  C )
371, 2, 3, 4, 30, 33, 36syl132anc 1202 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  D  =/=  C )
3837necomd 2626 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  C  =/=  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   joincjn 14321   meetcmee 14322   Latclat 14394   Atomscatm 29429   HLchlt 29516   LHypclh 30149
This theorem is referenced by:  cdleme11l  30434
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-psubsp 29668  df-pmap 29669  df-padd 29961  df-lhyp 30153
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