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Theorem cdleme10 33252
Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114.  D represents s2. In their notation, we prove s  \/ s2 = s  \/ r. (Contributed by NM, 9-Jun-2012.)
Hypotheses
Ref Expression
cdleme10.l  |-  .<_  =  ( le `  K )
cdleme10.j  |-  .\/  =  ( join `  K )
cdleme10.m  |-  ./\  =  ( meet `  K )
cdleme10.a  |-  A  =  ( Atoms `  K )
cdleme10.h  |-  H  =  ( LHyp `  K
)
cdleme10.d  |-  D  =  ( ( R  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  D )  =  ( S  .\/  R ) )

Proof of Theorem cdleme10
StepHypRef Expression
1 cdleme10.d . . 3  |-  D  =  ( ( R  .\/  S )  ./\  W )
21oveq2i 6288 . 2  |-  ( S 
.\/  D )  =  ( S  .\/  (
( R  .\/  S
)  ./\  W )
)
3 simp1l 1021 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  HL )
4 simp3l 1025 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  S  e.  A )
5 simp2 998 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  e.  A )
6 eqid 2402 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme10.j . . . . . 6  |-  .\/  =  ( join `  K )
8 cdleme10.a . . . . . 6  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 32364 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
103, 5, 4, 9syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  e.  (
Base `  K )
)
11 simp1r 1022 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  H )
12 cdleme10.h . . . . . 6  |-  H  =  ( LHyp `  K
)
136, 12lhpbase 32995 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1411, 13syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  ( Base `  K )
)
15 hllat 32361 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
163, 15syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  Lat )
176, 8atbase 32287 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
18173ad2ant2 1019 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  e.  ( Base `  K )
)
196, 8atbase 32287 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
204, 19syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  S  e.  ( Base `  K )
)
21 cdleme10.l . . . . . 6  |-  .<_  =  ( le `  K )
226, 21, 7latlej2 16013 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  S  .<_  ( R  .\/  S
) )
2316, 18, 20, 22syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  S  .<_  ( R  .\/  S ) )
24 cdleme10.m . . . . 5  |-  ./\  =  ( meet `  K )
256, 21, 7, 24, 8atmod3i1 32861 . . . 4  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  ( R  .\/  S
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  S  .<_  ( R  .\/  S
) )  ->  ( S  .\/  ( ( R 
.\/  S )  ./\  W ) )  =  ( ( R  .\/  S
)  ./\  ( S  .\/  W ) ) )
263, 4, 10, 14, 23, 25syl131anc 1243 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  ( ( R  .\/  S )  ./\  W )
)  =  ( ( R  .\/  S ) 
./\  ( S  .\/  W ) ) )
276, 7latjcom 16011 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  ( R  .\/  S )  =  ( S  .\/  R
) )
2816, 18, 20, 27syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  =  ( S  .\/  R ) )
29 eqid 2402 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
3021, 7, 29, 8, 12lhpjat2 33018 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  -> 
( S  .\/  W
)  =  ( 1.
`  K ) )
31303adant2 1016 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  W )  =  ( 1. `  K ) )
3228, 31oveq12d 6295 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( R  .\/  S )  ./\  ( S  .\/  W ) )  =  ( ( S  .\/  R ) 
./\  ( 1. `  K ) ) )
33 hlol 32359 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
343, 33syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  OL )
356, 7latjcl 16003 . . . . 5  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( S  .\/  R )  e.  ( Base `  K
) )
3616, 20, 18, 35syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  R )  e.  (
Base `  K )
)
376, 24, 29olm11 32225 . . . 4  |-  ( ( K  e.  OL  /\  ( S  .\/  R )  e.  ( Base `  K
) )  ->  (
( S  .\/  R
)  ./\  ( 1. `  K ) )  =  ( S  .\/  R
) )
3834, 36, 37syl2anc 659 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( S  .\/  R )  ./\  ( 1. `  K ) )  =  ( S 
.\/  R ) )
3926, 32, 383eqtrd 2447 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  ( ( R  .\/  S )  ./\  W )
)  =  ( S 
.\/  R ) )
402, 39syl5eq 2455 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  D )  =  ( S  .\/  R ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   lecple 14914   joincjn 15895   meetcmee 15896   1.cp1 15990   Latclat 15997   OLcol 32172   Atomscatm 32261   HLchlt 32348   LHypclh 32981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985
This theorem is referenced by:  cdleme10tN  33256  cdleme20aN  33308  cdleme20g  33314
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