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Theorem cdleme2 29568
Description: Part of proof of Lemma E in [Crawley] p. 113. .  F represents f(r).  W is the fiducial co-atom (hyperplane) w. Here we show that (r  \/ f(r))  /\ w = u in their notation (4th line from bottom on p. 113). (Contributed by NM, 5-Jun-2012.)
Hypotheses
Ref Expression
cdleme1.l  |-  .<_  =  ( le `  K )
cdleme1.j  |-  .\/  =  ( join `  K )
cdleme1.m  |-  ./\  =  ( meet `  K )
cdleme1.a  |-  A  =  ( Atoms `  K )
cdleme1.h  |-  H  =  ( LHyp `  K
)
cdleme1.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme1.f  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
Assertion
Ref Expression
cdleme2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( R  .\/  F )  ./\  W )  =  U )

Proof of Theorem cdleme2
StepHypRef Expression
1 cdleme1.l . . . 4  |-  .<_  =  ( le `  K )
2 cdleme1.j . . . 4  |-  .\/  =  ( join `  K )
3 cdleme1.m . . . 4  |-  ./\  =  ( meet `  K )
4 cdleme1.a . . . 4  |-  A  =  ( Atoms `  K )
5 cdleme1.h . . . 4  |-  H  =  ( LHyp `  K
)
6 cdleme1.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
7 cdleme1.f . . . 4  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
81, 2, 3, 4, 5, 6, 7cdleme1 29567 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  F )  =  ( R  .\/  U ) )
98oveq1d 5793 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( R  .\/  F )  ./\  W )  =  ( ( R  .\/  U ) 
./\  W ) )
10 simpll 733 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  HL )
11 simpr3l 1021 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  e.  A )
12 hllat 28704 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
1312ad2antrr 709 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  Lat )
14 simpr1 966 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  e.  A )
15 eqid 2256 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1615, 4atbase 28630 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1714, 16syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  e.  ( Base `  K )
)
18 simpr2 967 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  Q  e.  A )
1915, 4atbase 28630 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2018, 19syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  Q  e.  ( Base `  K )
)
2115, 2latjcl 14104 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
2213, 17, 20, 21syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
2315, 5lhpbase 29338 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2423ad2antlr 710 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  W  e.  ( Base `  K )
)
2515, 3latmcl 14105 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
2613, 22, 24, 25syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
276, 26syl5eqel 2340 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  e.  ( Base `  K )
)
2815, 1, 3latmle2 14131 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
2913, 22, 24, 28syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
306, 29syl5eqbr 4016 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  .<_  W )
3115, 1, 2, 3, 4atmod4i2 29207 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  U  e.  ( Base `  K )  /\  W  e.  ( Base `  K ) )  /\  U  .<_  W )  -> 
( ( R  ./\  W )  .\/  U )  =  ( ( R 
.\/  U )  ./\  W ) )
3210, 11, 27, 24, 30, 31syl131anc 1200 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( R  ./\  W )  .\/  U )  =  ( ( R  .\/  U ) 
./\  W ) )
33 eqid 2256 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
341, 3, 33, 4, 5lhpmat 29370 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  ./\  W
)  =  ( 0.
`  K ) )
35343ad2antr3 1127 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  ./\ 
W )  =  ( 0. `  K ) )
3635oveq1d 5793 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( R  ./\  W )  .\/  U )  =  ( ( 0. `  K ) 
.\/  U ) )
37 hlol 28702 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
3837ad2antrr 709 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  OL )
3915, 2, 33olj02 28567 . . . 4  |-  ( ( K  e.  OL  /\  U  e.  ( Base `  K ) )  -> 
( ( 0. `  K )  .\/  U
)  =  U )
4038, 27, 39syl2anc 645 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( 0. `  K )  .\/  U )  =  U )
4136, 40eqtrd 2288 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( R  ./\  W )  .\/  U )  =  U )
429, 32, 413eqtr2d 2294 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( R  .\/  F )  ./\  W )  =  U )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   Basecbs 13096   lecple 13163   joincjn 14026   meetcmee 14027   0.cp0 14091   Latclat 14099   OLcol 28515   Atomscatm 28604   HLchlt 28691   LHypclh 29324
This theorem is referenced by:  cdleme3  29577  cdleme37m  29802  cdleme39a  29805  cdleme50trn1  29889
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 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328
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